Basics On Logic

PROPOSITIONS A PROPOSITION is a statement that is either true (T) or false (F). Use letters p, q, r, P, Q, etc. to represent them. The negation of proposition P is denoted ~P Examples: P: Today is Monday ~P: Today is not Monday Q: There is a quiz next Friday ~Q: There will not be a quiz next Friday R: 2 is an odd number ~R: 2 is not an odd number (or 2 is an even number)

COMPOUND PROPOSITIONS P, Q atomic propositions New propositions P or Q P and Q If P then Q P if and only if Q (general form of a definition)

Truth-Value of Compound Propositions Proposition P or Q is FALSE when both P and Q are false otherwise true   Proposition P and Q is TRUE when both P and Q are true, otherwise false Proposition if P then Q is false only when P is true and Q is false (from a true fact you can not derive a false result). In any other case it is true. P is called the condition and Q the conclusion. If P then Q is the same as saying P is sufficient for Q, or also, Q is necessary for P. P if and only if Q is true when P and Q are true at the same time or P and Q are false at the same time.

EXAMPLE Determine the truth-value of JFK was president of the USA in 1959 or 1960 Paul is a CS major and has to take biology and chemistry If 3 is an even number, then 2 is even

Exercise 1 Determine the truth-value of each of the following statements. To accomplish it identify the atomic statements, determine their true value, and finally use the connectors involved in the statement to determine the truth-value of the compound statement a. I am a student at TAMUCC and I am a science major b. 3 divides 6 or 3 divides 10 c. If Δ is a square, Δ is a triangle

Exercise 2 In this exercise you are given compound statements and their true-value. Identify the connector(s) in the statement, the atomic propositions participating in the compound proposition, and the possible truth-values of the atomic propositions. Feel free to label the atomic propositions using letters. a. Either John wins the lottery or he finds a job (FALSE) c. Ruth is majoring in math and biology. (FALSE) e. The equation 2x - 2 = x2 - x has zero, one, or two real solutions. (TRUE)

EXERCISE 3 It was said that a statement of the form if P THEN Q is false only when P is true and Q is false (from a true fact you can’t derive a false conclusion). Consider the situation below to convince yourself about this claim. “The prize for the Texas lotto this week is $50 millions. Smart Janet decided to buy a ticket and made the following promise to his classmates: “If I win the lottery, I will take all of you to the Bahamas with all expenses paid” Discuss for which of the following scenarios her statement is true (she told the truth or fulfilled her promise), and for which it is false (she lied or did not fulfill her promise). • Janet won the lottery and took all the classmates to the Bahamas • Janet won the lottery but did not fulfill her promise • Janet did not win the lottery but took her classmates to the Bahamas • Janet did not win the lottery and did not take her classmates to the Bahamas

EXERCISE 4 Determine the truth-value of each of the following implications If 2 is negative, - 4 is positive   If 4 is even, 3 is even If 3 is odd, 4 is odd

EXERCISE 5 If the arrow is green, you have to turn left. (FALSE) For each of the implications below identify the condition and the conclusion. The truth-value of the implication is given (in parenthesis). Use the truth-value of the implication to determine the truth-value of the condition, and the truth-value of the conclusion. Sometimes your answer may have several choices of truth-values. If the arrow is green, you have to turn left. (FALSE) If today is cloudy then it will rain (TRUE) If Rob is taking calculus I at TAMU-CC, Rob is either a math major or a CS major (FALSE) The Music Appreciation class is offered only for students who are majoring in Music and Science (TRUE) If Chechi is from Mars, he is smart (FALSE)

STATEMENTS WITH “SOME”, “ALL” Statement may refer to a particular object in a set, to some of them or to all of them. For instance the statement below refer to the set of students in calculus I. Jacob is a student in calculus I (Is Jacob enrolled in the calculus I class?) Some calculus I are Computer Science majors How many? All students in Calculus I are Engineering majors When a statement refers to all the objects in a collection of objects (a set) the statement is said to be a universally quantified statement. When a statement refers to some of the objects in a collection (at least one) the statement is said to be an existentially quantified statement.

Universal Quantified Statements Science students at TAMUCC have to take calculus I. Other ways to express the same statement are: Any science student at TAMUCC has to take calculus I All science students at TAMUCC have to take calculus I Every (each) science student at TAMUCC has to take calculus I If you are a science student at TAMUCC then you have to take calculus I. It is a universally quantified statement for the collection of science students at TAMUCC. There are two collections (sets) involved in this discussion. One is the set of all science students at TAMUCC; the other one is the set of calculus students at TAMUCC. The quantifier is applied only to the collection of science majors, which is called the universal set for the statement. The statement is true only if each science student takes calculus I. Otherwise it is false.

Venn Diagram Calculus I students Science Students Any science student at TAMUCC takes calculus I Contrapositive of If you don’t take calculus I, you are not a science student

Existential Quantified Statements Some science students at TAMUCC take Greek. Other ways to express the same statement are: There is a science student at TAMUCC who takes Greek At least one science student at TAMUCC takes Greek   It is an existentially quantified for the set of science students. It refers only to some of the science students at TAMUCC. In this case you are referring to objects in two sets. One is the set of TAMUCC students; the other set is the set of students taking Greek. The statement is true if there is at least one science student who takes Greek.

Venn Diagram Greek students Science Students Some science students at TAMUCC take Greek

EXERCISE 6 Write the contrapositive for each of the following implications Any basketball player is taller than 5 feet. No calculus student is taking the karate class

EXERCISE 7 Indicate whether each of the following statements is existentially or universally quantified. Then determine their truth-value. Each student in the class wears glasses There is a student in the class who is at least 7 ft. tall Some students in the class are females All students are not classified as freshmen.

Exercise 8 This exercise deals with the negation of “for some” and “ for all” using Venn Diagrams. The statement below the diagram on each row describes the Venn diagram. The second diagram represents the negation of the first one. Your task is to write the proposition for the second Venn diagram and convince yourself that statement is the negation of the one to its left.

Exercise 9 Draw the Venn Diagrams for each of the statements below. Then determine the truth-value of each of them. If x is a real number different than zero, it has a multiplicative inverse. d. Any integer number is odd. g. Any science major takes a math course, but some students taking a math course are not science majors.

COUNTER EXAMPLES From our previous discussions it follows that the negation of “Any calculus I student wears glasses” Is “ There is at least one calculus student who does not wear glasses”.   To guarantee that “Any calculus I student wears glasses” is false, it ha to be guaranteed that “ There is at least one calculus student who does not wear glasses” is true. To do this it is enough to show one student in the class that does not wear glasses, for instance Lisa. Lisa does not wear glasses is a counterexample to show that “Any calculus I student wears glasses” is false.

EXERCISE 10 Give a counterexample to show that each of the statements below is false. a. Any TAMUCC is from Texas b. All numbers which are multiples of 3 are multiples of 6 c. All calculus I students at TAMUCC are science majors d. Any NBA player is at least 6 feet tall.