CLAST Logic.

CLAST Logic

What to Expect From CLAST Logic
There are 8 competencies, from which there will probably be 7 questions. You do not know on which competencies you will be tested. Plus, on the CLAST, there are sometimes experimental questions, which are not scored. created by Matt Kulmacz

created by Matt Kulmacz
Logic Symbols Ù and Ú or ® if...then º is equivalent to Ø not created by Matt Kulmacz

Statements A statement in logic takes the form: It is raining. John likes to go to dance class. Mary has a pet frog. The pot is hot. A statement is a sentence that asserts that something is true. There must be a subject, a verb, and some description or object or property. To simplify the identification of related parts of statements in an argument, we represent those parts as letters p, q, r, and s. EXAMPLE: The pot is hot. p is h Therefore, in logic, it is very important to “watch your p’s and q’s” created by Matt Kulmacz

Eight Logic Competencies Covered by CLAST
Venn Diagrams Negations of Statements Equivalence or Non-Equivalence of Statements Transforming Statements Without Affecting Their Meaning Drawing Conclusions From Data Identifying Invalid Arguments With True Conclusions Valid Reasoning Patterns Logical Conclusions When Facts Warrant Them Practice Test created by Matt Kulmacz

Competency 1: Venn Diagrams
U D E H M G I F U means “the universal set”, or all of the rectangle. D, E, F are 3 sets. Notice that the circles overlap, forming subsets G, H, I, and M, for a total of eight separate areas. The area within the box, but outside the circles, counts! created by Matt Kulmacz

Example 1 U D E F Assuming none of the regions are empty, which of the following statements is true: (A) Any element which is a member of set E is also a member of set D. (B) No element is a member of all 3 sets, D, E, and F. (C) Any element which is a member of set U is also a member of set D. (D) None of the above statements is true. created by Matt Kulmacz

Example 1 Choice A U D E F (A) Any element which is a member of set E is also a member of set D. false If anything which is an E is also a D, then the portion of the yellow circle, above, in which the E appears can have no members --- they would have to be inside the green circle as well, to be a D. But that portion of circle E is not empty (no region is empty). Therefore there must be some member which is an E which is not a D. created by Matt Kulmacz

Example 1 Choice B U D E F (B) No element is a member of all 3 sets, D, E, and F. false The reasoning here is similar, but more direct. A thing which is a D, an E, and also an F at the same time would be represented by a mark within the little area in the middle, the one that looks like a triangle with slightly curved sides. Statement (B) says that this area must be empty, because there is no such member. But no region is empty! created by Matt Kulmacz

Example 1 Choice C U D E F (C) Any element which is a member of set U is also a member of set D false A member of U is anything within the rectangle. Statement (C) claims that anything inside the rectangle U must also be within the circle D. That would mean that the portions of circles E and F that are outside circle D have no members, and are empty. But no region is empty! Therefore there are members of E (and F) that are members of U (because they are inside the rectangle) that are not members of D. created by Matt Kulmacz

Example 1 Summary U D E F (A) Any element which is a member of set E is also a member of set D. false (B) No element is a member of all 3 sets, D, E, and F. (C) Any element which is a member of set U is also a member of set D. (D) None of the above statements is true. true, since each of (A), (B), and (C) are false. created by Matt Kulmacz

Example 2 U J K L Assuming none of the regions are empty, which of the following statements is true: (A) Any element which is a member of set U is also a member of set J. (B) No element is a member of both sets K and L. (C) No element is a member of both sets J and K. (D) None of the above statements is true. created by Matt Kulmacz

Example 2 Answer U J K L (A) Any element which is a member of set U is also a member of set J. false (B) No element is a member of both sets K and L. (C) No element is a member of both sets J and K. true (D) None of the above statements is true. Back to competencies created by Matt Kulmacz

Competency 2: Negations of Statements
Take a moment to write down this table: Note: you should be able to translate these phrases into the language of logic using the symbols introduced earlier. not ( not p ) created by Matt Kulmacz

Examples: Statement: It is raining. Negation: It is not raining. Statement: It is not raining. Negation: It is raining. Statement: It is raining outside and the sun is shining. Negation: It is not raining outside or the sun is not shining. Statement: Pat plays baseball or he plays football. Negation: Pat does not play baseball and he does not play football. Statement: If it snows then I will be cold. Negation: It snows and I am not cold. created by Matt Kulmacz

Negating the “NOT” statement
Using symbolic logic: Ø (Ø p ) º p Using words: not (not p) is equivalent to p Negating: it is not raining outside. is equivalent to: it is raining outside. created by Matt Kulmacz

Negating the “AND” statement
Using symbolic logic: Ø (p  q) º ( p   q) Using words: not (p and q) is equivalent to (not p or not q) Negating: it is raining outside and the sun is shining. is equivalent to: it is not raining outside or the sun is not shining. created by Matt Kulmacz

Negating the “OR” statement
Using symbolic logic: Ø (p  q) º ( p   q) Using words: not (p or q) is equivalent to (not p and not q) Negating: it is raining outside or the sun is shining. is equivalent to: it is not raining outside and the sun is not shining. created by Matt Kulmacz

Negating Compound Statements
A compound statement is two simple statements joined by the word AND or the word OR. Negating a compound statement requires that you negate both simple statements. Also you must exchange which word is used to join them. Ø (p  q) º ( p   q) not (It is raining outside and the sun is shining). becomes It is not raining outside or the sun is not shining. Ø (p  q) º ( p   q) not (It is raining outside or the sun is shining). becomes It is not raining outside and the sun is not shining. These two rules of logic are called De Morgan’s Laws after the mathematician who proved them for the first time, Augustus De Morgan. It might help you to remember them to give them a name. created by Matt Kulmacz

The tricky negation: the IF statement
Using symbolic logic: Ø (p  q) º ( p   q) Using words: not (if p then q) is equivalent to (p and not q) not (If it rains then there are clouds) is equivalent to: it rains and there are no clouds. Memorize this pattern of negation. It is too hard to “think out”. created by Matt Kulmacz

The tricky negation: the IF statement 2
Using symbolic logic: Ø (p  q) º ( p   q) Using words: not (if p then q) is equivalent to (p and not q) If p then q is equivalent to (not p) or q Thus not (If p then q) is equivalent to not ((not p) or q) which by one of De Morgan’s laws is p and (not q) created by Matt Kulmacz

Your Turn Write the negation of the following statements: It is cold and I will turn the heater on. If it rains then the boys will go to the mall. I will write a paper or I will not finish this class. If I play table tennis, then I always win I am going to the mountains and I will see snow. Joe has a headache or he is playing basketball. It is not snowing. created by Matt Kulmacz

Answers (negation of statements) 1
It is cold and I will turn the heater on. It is not cold or I will not turn the heater on. If it rains then the boys will go to the mall. It rains and the boys will not go to the mall. I will write a paper or I will not pass this class. I will not write a paper and I will pass this class. Notice two not’s cancel. created by Matt Kulmacz

Answers (negation of statements) 2
If I play table tennis, then I always win I play table tennis and I do not always win. I am going to the mountains and I will see snow. I am not going to the mountains or I will not see snow Joe has a headache or he is playing basketball. Joe does not have a headache and he is not playing basketball. It is not snowing. It is snowing. created by Matt Kulmacz

Negating statements that contain quantifiers:
A quantifier is a word that tells “how many”. All, Some, Some-not, and No are quantifiers. created by Matt Kulmacz

the “Square of Opposition”
Some are All Negation Negation Some are not No Copy and memorize!!! You may find it easier to memorize the negations as below Some are – No Some are not – All created by Matt Kulmacz

Some are All Negation Negation Some are not No Examples: Statement: Some fish do not have skin. No roses are red. All pansies are multicolored. Some students do not take history. Some cats have blue eyes. Some trees have green leaves. Negation All fish have skin. Some roses are red. Some pansies are not multicolored. All students take history. No cat has blue eyes. No trees have green leaves. created by Matt Kulmacz

Now Your Turn Negate the following statements: (answer orally) Some dogs do not eat meat. No roses are red. Some cats have nine lives. All men are mortal. created by Matt Kulmacz

Answers The negations of the statements: Some dogs do not eat meat. ans: All dogs eat meat. No roses are red. ans: Some roses are red. Some cats have nine lives. ans: No cats have nine lives. All men are mortal. ans: Some men are not mortal. Back to competencies created by Matt Kulmacz

Competency 3: Equivalence or Non-Equivalence of Statements
In the previous section we discussed the AND, OR , and IF statements and what these statements become when negated. So, we have covered equivalencies of negated statements. created by Matt Kulmacz

Statement Equivalence Rules
In symbolic logic: In word form:  (p  q)  ( p   q) not (p and q)  (not p) or (not q)  (p  q)  ( p   q) not (p or q)  (not p) and (not q)  (p  q)  (p   q) not (if p then q)  p and (not q) Some people find it easier to remember the equivalence rules in symbolic logic form; some find the word forms easier to remember. The first and second rules above are De Morgan’s Rules. created by Matt Kulmacz

“if...then” equivalencies
Copy these equivalencies: In symbolic logic: if p then q is equivalent to (not p) or q In words: if p then q is equivalent to if (not q) then (not p) created by Matt Kulmacz

Caution on “if…then” statements
Don’t mix the negation of the if...then statement with the equivalencies for the if...then statement. created by Matt Kulmacz

Example of “if...then” equivalencies
If I live in Fort Myers, then I live in Florida. is equivalent to: (1) I do not live in Fort Myers or I live in Florida. (2) If I do not live in Florida then I do not live in Ft. Myers. created by Matt Kulmacz

Now Your Turn Write the equivalencies of: If I have a hat, then I have a warm head. created by Matt Kulmacz

Your Turn Answer Write the equivalencies of: If I have a hat, then I have a warm head. if h then w Answers: (1) I do not have a hat or I have a warm head. (not h) or w (2) If I do not have a warm head then I do not have a hat. if (not w) then (not h) Note: the equivalencies of the “If” statement are about as hard as it gets. So don’t panic at this point in the tutorial!!! created by Matt Kulmacz

A little trick: You may be asked what is NOT EQUIVALENT. For “If p then q” look for the following forms. If q then p This is the converse of the original, if p then q. If (not p) then (not q) This is the converse of the correct equivalency, if (not q) then (not p). created by Matt Kulmacz

You try: Write 2 statements equivalent to: If Jo passes algebra, then she will enroll in geometry. created by Matt Kulmacz

You try: Answers Write 2 statements equivalent to: If Jo passes algebra, then she will enroll in geometry. p q Ans 1: if not q then not p If Jo will not enroll in geometry then she did not pass algebra. Ans 2: (not p) or (q) Jo does not pass algebra or she will enroll in geometry. not q not p not p q created by Matt Kulmacz

Another One for You Write 2 statements NOT equivalent to: If Jo passes algebra, then she will enroll in geometry. ans 1: if q then p If Jo enrolls in geometry then she passed algebra. ans 2: if not p then not q If Jo does not pass algebra then she will not enroll in geometry. q p not p not q created by Matt Kulmacz

Sample Question Here’s a type of question you may find on the CLAST. Find the equivalence of: It is not true that both Bill and Tom are students. created by Matt Kulmacz

Solution This is equivalent to a negated “and” statement: not (p and q)  not p or not q It is not true that both Bill and Tom are students. Rephrased as: It is not true that Bill is a student and Tom is a student. p q Thus, the equivalence is: Bill is not a student or Tom is not a student. not p not q created by Matt Kulmacz

Rephrasing Statements
Remember, if you see a statement prefaced by either It is not true that both... or It is true that both... You may want to rephrase the statement by eliminating the word both before you apply the rules of equivalency. It is not true that both Mary and Alice are happy. rephrased It is not true that Mary is happy and Alice is happy. It is not true that both Mary or Alice are happy. It is not true that Mary is happy or Alice is happy. Back to competencies created by Matt Kulmacz

Competency 4: Identifying Rules for Transforming Statements
Statements may be transformed (rewritten) without affecting their meaning. The next slide provides a table illustrating the rules for transforming statements. created by Matt Kulmacz

Rules for Transforming Statements
Equivalent Statement not (not p) p not (p and q) (not p) or (not q) not (p or q) (not p) and (not q) if p then q (not p) or (q) not all are p some are not p all are not p none are p not (if p, then q) (p) and (not q) if not q then not p You have seen these before, so this competency should be a review. created by Matt Kulmacz

Example 1 Write the rule that transforms statement “1” into statement “2”. 1. If today is Saturday, then Joe will play golf. 2. Today is not Saturday or Joe plays golf. Form for 1: if p, then q Form for 2: (not p) or (q) Solution: “if p, then q” is equivalent to “(not p) or (q)”. created by Matt Kulmacz

Example 2 Write the rule that transforms statement “1” into statement “2”. 1. Not all people cook. 2. Some people do not cook. Form for 1: not all are p Form for 2: some are not p Solution: “not all are p” is equivalent to “some are not p”. created by Matt Kulmacz

Example 3 Write the rule that transforms statement “1” into statement “2”. 1. It is not true that it is raining and Susan is playing golf. 2. It is not raining or Susan is not playing golf. Form for 1: not (p and q) Form for 2: (not p) or (not q) Solution: “not (p and q)” is equivalent to “(not p) or (not q)”. created by Matt Kulmacz

Applying Transformations
Write the statements that illustrate the rule “If p then q” is equivalent to “if not q then not p” using p to represent “The temperature is 30 degrees” and q to represent “Hyo will wear a coat”. If the temperature is 30 degrees then Hyo will wear a coat. is equivalent to: If Hyo does not wear a coat, then the temperature is not 30 degrees. created by Matt Kulmacz

Example 4 Write the rule that transforms statement “1” into statement “2”. 1. It is not true that Tom is sleeping and the alarm is ringing. 2. Tom is not sleeping or the alarm is not ringing. Form for 1: not (p and q) Form for 2: (not p) or (not q) Solution: “not (p and q)” is equivalent to “(not p) or (not q)”. created by Matt Kulmacz

Example 5 Write the rule that transforms statement “1” into statement “2” in one step (this is the phraseology used on the CLAST). 1. If 3x is even then x is even. 2. If x is not even, then 3x is not even. “If p then q” is equivalent to “If not q, then not p ”. created by Matt Kulmacz

Example 6 Write the rule that transforms statement “1” into statement “2” in one step. 1. Not all of the men and women are intelligent. 2. Some man or woman is not intelligent. “Not all are p” is equivalent to “Some are not p”. Back to competencies created by Matt Kulmacz

Competency 5: Drawing Conclusions From Data
Form: Two premises will be given, and from them we are to draw a logical conclusion, following the templates on the next slide. created by Matt Kulmacz

Patterns of Argumentation
Given that It rains or the sun shines It does not rain The sun shines p or q not p Conclusion q The sun does not shine It rains not q Conclusion p If it rains, then the sun shines if p then q p Conclusion not p If the sun shines, then I go to the beach. If it rains, then I go to the beach. if q then r Conclusion if p then r Patterns of Argumentation Note the similarity between the first two patterns. We need only state one of these forms to get the understanding. created by Matt Kulmacz

the Basic Patterns Memorize these four patterns. created by Matt Kulmacz

Understanding the Basic Patterns
p is true, q is true, or both are true. p is not true. Therefore q must be true. q is a consequence of p. p happens. Therefore q must happen. q is a consequence of p. q doesn’t happen. Therefore p must not have happened. q is a consequence of p. r is a consequence of q. Therefore r must be a consequence of p. created by Matt Kulmacz

Argument 1 Determine the argument and what can be deduced. Joe goes to a movie or he goes to the beach. Joe does not go to the beach. Answer: pattern 1 Joe goes to a movie. created by Matt Kulmacz

Argument 2 Determine the argument and what can be deduced. If Maria likes math, then she will become an engineer. Maria likes math. Answer: pattern 2 Maria will become an engineer. created by Matt Kulmacz

Argument 3 Determine the argument and what can be deduced. If Emmett sings, then he will not play the piano. Emmett plays the piano. Answer: pattern 3 Emmett does not sing. created by Matt Kulmacz

Argument 4 Determine the argument and what can be deduced. If it rains, I will go to the movies. If I go to the movies then I don’t clean my room. Answer: pattern 4 If it rains then I don’t clean my room. created by Matt Kulmacz

Rephrasing Premises No people are happy. All girls are people. The premises as given do not match any of the four basic patterns, so you must rephrase them. If you are a person then you are not happy. If you are a girl then you are a person. Answer: pattern 4 If you are a girl, then you are not happy. (OR: No girls are happy.) Back to competencies created by Matt Kulmacz

Competency 6: Identifying invalid arguments with true conclusions
We will be talking about 2 premises (assumptions or observations) and a conclusion. You need to be able to recognize an argument where both premises hold, but the conclusion does not necessarily follow. created by Matt Kulmacz

Example 1 premise: All integers are real numbers. premise: All rational numbers are real numbers. conclusion: Therefore all integers are rational numbers. This is an invalid argument even though the conclusion is true. The best way to determine this is by Venn diagrams. created by Matt Kulmacz

Venn Diagram of Example 1
Real Integers Premise 1: All integers are real numbers Premise 2: All rational numbers are real numbers Real Real Real Real Integers Rational Integers Integers Integers Rational Rational Rational Conclusion: Because there are several diagrams that support the premises, you are not forced to accept the conclusion that all integers are rational numbers. Therefore, this is an invalid argument with a true conclusion. created by Matt Kulmacz

Remember: If you can draw several diagrams that support the premises, you are not forced to accept the conclusion. Thus, the argument is invalid even though the conclusion is true. created by Matt Kulmacz

Warning: Don’t let any knowledge that you know interfere with your analysis of the logical argument. Deal only with the two premises and the conclusion and what they contain. You must understand that it is NOT the truth or falseness of the conclusion that is in question. It is whether we are justified in making that conclusion on the basis of the information provided by the premises. created by Matt Kulmacz

Try this one premise: All rational numbers are real numbers. premise: 17 is a rational number. conclusion: Therefore, 17 is a real number. The conclusion is true, but is the argument? created by Matt Kulmacz

Venn Diagram Solution Real Premise 1: All rational numbers are real numbers Rational Real Rational Premise 2: 17 is a rational number 17 Conclusion: Because there is only one diagram that supports the premises, you are forced to accept the conclusion that 17 is a real number. Therefore, the argument is valid. created by Matt Kulmacz

Another One for You premise: All dogs are animals. premise: All poodles are animals. conclusion: Therefore, poodles are dogs. Is this an invalid argument with a true conclusion? created by Matt Kulmacz

Answer Animals Dogs Premise 1: All dogs are animals. Premise 2: All poodles are animals. Animals Animals Animals Animals Dogs Dogs Poodles Dogs Dogs Poodles Poodles Poodles Conclusion: Because there are several diagrams that support the premises, you are not forced to accept the conclusion that all poodles are dogs. Therefore, this is an invalid argument with a true conclusion. Back to competencies created by Matt Kulmacz

Competency 7: Valid Reasoning Patterns
You may be asked to determine a conclusion, given two premises. This can be done if you know the logical reasoning pattern, given the premises This section is very similar to section 5: drawing conclusions from data. Although it is a bit harder than section 5, you should find this section a review. Refer to your notes from section 5. created by Matt Kulmacz

Patterns of Reasoning I play the piano or the violin I don’t play the piano I play the violin If I am sick then I must take medicine I am sick I must take medicine If I work hard then I will get a promotion I did not get a promotion I did not work hard If I attend school, then I will graduate If I graduate, then I will get a job. If I attend school then I will get a job created by Matt Kulmacz

Now, you try Determine the pattern and express the conclusion: If all students study, then no failing grades are given. Some failing grades are given. created by Matt Kulmacz

Solution if p then q not q not p If all students study,
then no failing grades are given. Some failing grades are given. Pattern: if p then q not q not p p q not q (Remember the negation of “no” is “some”) (Remember, the negation of “all” is “some-not”) Conclusion: Some students do not study. not p created by Matt Kulmacz

Try again Determine the pattern and express the conclusion: If no person passes the test, then some questions were unfair. If some questions were unfair then the teacher should be fired. created by Matt Kulmacz

Solution If no person passes the test, then some questions were unfair. If some questions were unfair, then the teacher should be fired. Pattern: if p then q if q then r if p then r p q q r Conclusion: If no person passes the test, then the teacher should be fired. p r Back to competencies created by Matt Kulmacz

Competency 8 Logical Conclusions When Facts Warrant Them
Here is where it gets complicated. We will be putting it all together: logic problems with 2 to 4 premises; premises that may involve simple or compound statements; statements that contain negations or quantifiers. The conclusion must logically follow from these premises. created by Matt Kulmacz

Summary of Equivalent Statements
created by Matt Kulmacz

Patterns of Reasoning p or q not q p note: these are the same basic pattern. created by Matt Kulmacz

Example 1 If you go to a picnic, then you have a great time. If you have a great time then you feel good. You go to a picnic. If you assign symbols to the simple statements making up the premises, you can see the structure. Structure p: you go to a picnic P1: If p then g g: you have a great time P2: If g then f f: you feel good P3: p created by Matt Kulmacz

Example 1: first method, 1 Comparing the first two premises of the example to the patterns of reasoning, we see that there is the same basic structure as pattern 4: Pattern 4 Problem if p then q P1: If p then g if q then r P2: If g then f if p then r So we may make a preliminary conclusion: C1: If p then f In words: If you go to the picnic, then you (will) feel good. created by Matt Kulmacz

Example 1: first method, 2 Looking at the last premise combined with this new fact, the intermediate conclusion, we see that there is the same basic structure as pattern 2: Pattern 2 Problem if p then q C1: If p then f p P3: p q Now we may make a final conclusion: C2: f In words: You (will) feel good. created by Matt Kulmacz

Example 1: second method, 1
Comparing the first and last premises of the example to the patterns of reasoning, we see that there is the same basic structure as pattern 2: Pattern 2 Problem if p then q P1: If p then g p P3: p q So we may make a different preliminary conclusion: C3: g In words: You (will) have a great time. created by Matt Kulmacz

Example 1: second method, 2
Comparing the second premise combined with this new intermediate conclusion, we see that there is still the structure as pattern 2: Pattern 2 Problem if p then q P1: If g then f p P3: p q So we may make the same final conclusion: C2: f In words: You (will) feel good. (We had better be able to come to the same final conclusion!) created by Matt Kulmacz

Did you notice ... … from the preceding example, that drawing logical conclusions has a “snowball” effect? You can arrange premises in any order to optimize your use of the reasoning patterns. created by Matt Kulmacz

Example 2 Joe plays golf or Sam plays tennis. If Joe plays golf, then he owns golf clubs. Sam does not play tennis. Assign symbols to clarify the structure. Structure g: Joe plays golf P1: g or t t: Sam plays tennis P2: If g then c c: Joe owns golf clubs P3: not t created by Matt Kulmacz

Example 2: solution, 1 This example does not offer you the luxury of alternate analysis. Comparing the first and last premises of the example to the patterns of reasoning, we see that there is the same basic structure as pattern 1: Pattern 1 Problem p or q P1: g or t not q P3: not t p So we may make a preliminary conclusion: C1: g In words: Joe plays golf. created by Matt Kulmacz

Example 2: solution, 2 Looking at the second premise combined with the intermediate conclusion, we see that there is the same basic structure as pattern 2: Pattern 2 Problem if p then q P2: If g then c p C1: g p So we may make the final conclusion: C2: c In words: Joe owns golf clubs created by Matt Kulmacz

Sample Problem 1 Select the appropriate conclusion. If Joe is a music major, he plays the piano. If he plays the piano, he plays the organ. Joe plays the organ. A. Joe is not a music major. B. Joe does not play the piano. C. Joe is a music major. D. None of the above is warranted. created by Matt Kulmacz

Solution If Joe is a music major, he plays the piano. if m then p If he plays the piano, he plays the organ. if p then o Joe plays the organ. o pattern 4 if p then q if m then p if q then r if p then o if p then r C1: if m then o Now we have the following: if m then o o This is NOT one of the patterns of reasoning, so we may not conclude anything. The answer is “D”. created by Matt Kulmacz

Sample Problem 2 Select the appropriate conclusion. If it rains, then the ball game will not be played. If the ball game is not played, then the boys will go to a movie. The boys go to the mall or it rained. They didn’t go to the mall. A. It does not rain. B. The ball game will be played. C. The boys go to a movie. D. None of the above is warranted. created by Matt Kulmacz

Solution, part 1 If it rains, then the ball game will not be played if r then p If the ball game is not played, then the boys will go to a movie if p then o The boys go to the mall or it rained m or r They didn’t go to the mall not m pattern 4 if p then q if r then p if q then r if p then o if p then r C1: if r then o pattern 1 p or q m or r not p not m q C2: r created by Matt Kulmacz

Solution, part 2 Assigned variables: r: it rains p: the ball game will not be played o: the boys will go to a movie m: the boys go to the mall pattern 4 if p then q if r then o p r q C3: o The answer is “the boys will go to a movie, or “C”. Back to competencies created by Matt Kulmacz

Practice Test Write your answers on a separate piece of paper. Back to competencies created by Matt Kulmacz

1. Assuming none of the regions is empty, which of the following statements is true? U K L J A. Any element that is a member of set J is also a member of set K. B. No element is a member of all three sets: J, K, and L. C. Any element that is a member of set U is also a member of set J. D. None of the above statements is true. created by Matt Kulmacz

2. Assuming none of the regions is empty, which of the following statements is true? U Q F K A. Any element that is a member of set Q is also a member of set F. B. No element is a member of all three sets: F, Q, and K. C. Any element that is a member of set U is also a member of set Q. D. None of the above statements is true. created by Matt Kulmacz

3. Select the statement that is the negative of the statement “All winter days are cold.” A. Some winter days are cold. B. Some winter days are not cold. C. No winter days are cold. D. If it is not a winter day, then it is not cold. created by Matt Kulmacz

4. Select the statement that is the negative of the statement “The sun is shining or the store is closed.” A. The sun is shining or the store is not closed. B. The sun is shining and the store is not closed. C. The sun is not shining and the store is not closed. D. If the sun is shining, then the store is not closed. created by Matt Kulmacz

5. Select the statement that is the negative of the statement “If the weather is cold, then the ball game will not be played.” A. If the weather is not cold, then the ball game will be played. B. The weather is cold and the ball game was not played. C. If the ball game is played, then the weather is not cold. D. The weather is cold and the ball game will be played. created by Matt Kulmacz

6. Select the statement that is the negative of the statement “No cats are felines.” A. Some cats are felines. B. Some cats are not felines. C. If an animal is a feline, then it is not a cat. D. All cats are felines. created by Matt Kulmacz

7. Select the statement below that is logically equivalent to “If Tom studies, then he will pass CLAST.” A. If Tom does not study, then he will not pass CLAST. B. If Tom passed CLAST, then he studied. C. If Tom did not pass CLAST, then he did not study. D. Tom studies and does not pass CLAST. created by Matt Kulmacz

8. Select the statement below that is logically equivalent to “It is not true that Jim is playing golf or Mary is playing tennis.” A. Jim is not playing golf or Mary is playing tennis. B. Jim is playing golf and Mary is not playing tennis. C. If Jim is not playing golf, then Mary is not playing tennis. D. Jim is not playing golf and Mary is not playing tennis. created by Matt Kulmacz

9. Select the statement below that is NOT logically equivalent to “If Mary works late, then Joe will prepare dinner.” A. If Joe prepares dinner, then Mary works late. B. If Joe does not prepare dinner, then Mary did not work late. C. Mary does not work late or Joe prepares dinner. created by Matt Kulmacz

10. Select the statement below that is logically equivalent to “It is not true that some dogs bark or some birds do not have feathers.” A. Some dogs do not bark and some birds have feathers. B. No dogs bark and all birds have feathers. C. Some dogs do not bark or some birds have feathers. D. No dogs bark and some birds have feathers. created by Matt Kulmacz

11. Select the rule of logical equivalence that directly (in one step) transforms statement “i” into statement “ii”. i) If Joe takes calculus, then he will buy a calculator. ii) Joe will not take calculus or he will buy a calculator. A. “If p, then q” is equivalent to “if not q, then not p.” B. “If p, then q” is equivalent to “(not p) or q.” C. Not (p and q)” is equivalent to “(not p) or (not q).” D. Correct equivalence rule is not given. created by Matt Kulmacz

12. Select the rule of logical equivalence that directly (in one step) transforms statement “i” into statement “ii”. i) Not all of the students have calculators. ii) Some students do not have calculators. A. “If p, then q” is equivalent to “if not q, then not p.” B. “All are not p” is equivalent to “none are p.” C. “Not (not p)” is equivalent to “p.” D. “Not all are p” is equivalent to “some are not p”. created by Matt Kulmacz

13. Read the requirements and each applicant’s qualifications for the opportunity to buy a condominium at the beach. To qualify to buy a condominium an applicant must have a gross income of $50,000 if single ($65,000 combined income if married) and have no pets. Mr... Blue is single and makes $55,000. He has a cat. Mrs. Green and her husband have no pets. They both work. Mr... Green makes $32,000 and Mrs.. Green makes $35,000. Mr.. Pao is married. He makes $60,000. His wife has no income. They have no pets. Identify which applicant would qualify for the opportunity to buy a condominium. A. Mr.. Blue B. Mrs. Green C. Mr.. Pao D. None of the above created by Matt Kulmacz

14. Given that: i) No athletes are lazy ii) All basketball players are athletes. determine which conclusion can be logically deduced. A. No basketball player is lazy. B. All basketball players are lazy. C. Some basketball players are lazy. D. None of the above is true. created by Matt Kulmacz

15. All of the following arguments A-D have true conclusions, but one of the arguments is not valid. Select the argument that is not valid. A. All birds have wings and all robins are birds; therefore, all robins have wings. B. All robins have wings and all birds have wings; therefore, all robins are birds. C. All turtles are reptiles and all reptiles have a scaly skin; therefore, turtles have a scaly skin. D. All mammals have hair. A deer is a mammal. Therefore, a deer has hair. created by Matt Kulmacz

16. All of the following arguments A-D have true conclusions, but one of the arguments is not valid. Select the argument that is not valid. A. All spiders are predaceous. The black widow is predaceous. Therefore, the black widow is a spider. B. All sea stars are echinoderms and all echinoderms are marine; therefore, all sea stars are marine. C. All frogs are amphibians and all amphibians breathe by lungs, gills, or skin; therefore, all frogs breathe by lungs, gills or skin. D. all mammals have hair and all giraffes are mammals; therefore, all giraffes have hair. created by Matt Kulmacz

17. Select the conclusion that will make the following argument valid. i) If I pass the CLAST, then I will get my AA degree. ii) If I get my AA degree, then I will attend the university. A. If I do not pass the CLAST, then I will not attend the university. B. If I get my AA degree, then I pass the CLAST. C. If I pass the CLAST, then I will attend the university D. If I pass the CLAST, then I will not attend the university. created by Matt Kulmacz

18. Select the conclusion that will make the following argument valid. If I am tired, then I need more sleep. I do not need more sleep. A. If I need more sleep, then I am tired. B. If I am not tired, then I do not need more sleep. C. I am tired. D. I am not tired. created by Matt Kulmacz

19. Select the conclusion that will make the following argument valid. Some people vote or all issues pass. No people vote. A. Some issues did not pass B. All issues pass. C. All people do not vote and some issues do not pass. D. No people vote and some issues did not pass. created by Matt Kulmacz

20. Study the information given below. If a logical conclusion is given, select that conclusion. If none of the conclusions given is warranted, select option D. If I pass this test, then I will graduate. I pass this test or I get a job. I did not get a job. A. I did not pass this test. B. I did not graduate. C. I did graduate. D. None of the above is warranted. created by Matt Kulmacz

Study the information given below. If a logical conclusion is given, select that conclusion. If none of the conclusions given is warranted, select option D. If all cars have cruise control, then no person speeds. If no person speeds, then no tickets are given. Some tickets are given. 21. A. No person speeds. B. Some cars do not have cruise control. C. All cars have cruise control. D. None of the above is warranted. created by Matt Kulmacz

22. Study the information given below. If a logical conclusion is given, select that conclusion. If none of the conclusions given is warranted, select option D. Mary eats ice cream or she eats yogurt. If Mary eats yogurt, then she is healthy. If Mary is healthy, then she can run the marathon. Mary does not eat yogurt. A. Mary does not eat ice cream. B. Mary is healthy. C. If Mary runs the marathon, then she eats yogurt. D. None of the above is warranted. created by Matt Kulmacz

Answers: Venn Diagrams 1. B 2. D Transforming Statements Without Affecting Their Meaning 11. B 12. D Valid Reasoning Patterns 17. C 19. B 18. D Negations of Statements 3. B 5. D 4. C 6. A Logical Conclusions When Facts Warrant Them 20. C 22. D 21. B Drawing Conclusions From Data 13. B 14. A Equivalence or Non-Equivalence of Statements 7. C 9. A 8. D 10. B Identifying Invalid Arguments With True Conclusions 15. B 16. A Back to competencies created by Matt Kulmacz

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