Chapter 8 Logic DP Studies

Content A Propositions B Compound propositions C Truth tables and logical equivalence D Implication and equivalence E Converse, inverse, and contrapositive F Valid arguments b

Opening Problem b

A. Propositions Propositions are statements which may be true or false. Examples: 1. The sky is green. 2. There is a virus on this computer.

A. Propositions Questions are not propositions. Is the sky green today? Comments or opinions that are subjective, are also not propositions since they are not definitely true or false. ‘Green is a nice color’ Propositions may be indeterminate. ‘your father is 50 years old’ would not have the same answer (true or false) for all people.

A. Proposition The truth value of a proposition is whether it is true or false. example: 1. one plus one equals two ( T ) 2. one plus one equals three ( F )

A. Proposition Example 1

A. Proposition Solution to example 1

A. Proposition We represent propositions by letters such as p, q, and r. For example, p: It always rains on Tuesdays. q: = 46 r: x is an even number.

A. Proposition

We can also represent negation using a truth table. p TF FT

A. Proposition Example 2

A. Proposition Solutions to example 2:

A. Proposition We can use a Venn diagram to represent these propositions and their negations.

A. Proposition Example 3

A. Proposition Solutions to example 3:

B. Compound Propositions Compound propositions are statements which are formed using connectives such as and and or. For example: p: Bob went to the beach q: Bob flew a kite p ^ q: Bob went to the beach and flew a kite

B. Compound Propositions When two propositions are joined using the word and, the new proposition is the conjunction of the original propositions. If p and q are propositions, p ^ q is used to denote their conjunction. For example: p: Eli had soup for lunch q: Eli had a pie for lunch p ^ q: Eli had soup and a pie for lunch. A conjunction is true only when both original propositions are true.

B. Compound Propositions Conjunction truth table pqp ^ q TTT TFF FTF FFF p ^ q is true when both p and q are true. The first 2 columns list the possible combinations for p and q.. p ^ q is false whenever one or both of p and q are false.

B. Compound Propositions

When two propositions are joined by the word or, the new proposition is the disjunction of the original propositions. If p and q are propositions, p V q is used to denote their disjunction. For example: p: Frank played tennis today q: Frank played golf today p V q: Frank played tennis or golf today. p V q is true if Frank played tennis or golf or both today. A disjunction is true when one or both propositions are true. A disjunction is only false if both propositions are false.

B. Compound Propositions The truth table for the disjunction “p or q” is: pqp V q TTT TFT FTT FFF p V q is true if p or q or both are true. p V q is only false if both p and q are false.

B. Compound Propositions We can use Venn diagrams to represent disjunction.

B. Compound Propositions The exclusive disjunction is true when only one of the propositions is true. The exclusive disjunction of p and q is written p V q. We can describe p V q as “p or q, but not both”, or “exactly one of p and q”. For example: p: Sally ate cereal for breakfast q: Sally ate toast for breakfast p V q: Sally ate cereal or toast, but not both, for breakfast.

B. Compound Propositions The truth table for the exclusive disjunction p V q is: pqp V q TTF TFT FTT FFF p Y q is true if exactly one of p and q is true. p Y q is false if p and q are both true or both false.

B. Compound Propositions If P and Q are the truth sets for propositions p and q respectively, then the truth set for p V q is the region shaded, where exactly one of p and q is true.

B. Compound Propositions Example 4:

B. Compound Propositions Solutions to example 4:

C. Truth Tables and Logical Equivalence This is what we know so far.

C. Truth Tables and Logical Equivalence

Solution to example 5

C. Truth Tables and Logical Equivalence Solution to example 5

C. Truth Tables and Logical Equivalence Solution to example 5

C. Truth Tables and Logical Equivalence A compound proposition is a tautology if all the values in its truth table column are true. Example:

C. Truth Tables and Logical Equivalence

Solution to example 8:

C. Truth Tables and Logical Equivalence When three propositions are under consideration, we usually denote them p, q, and r. The possible combinations of the truth values for p, q, and r are listed systematically in the table alongside.

C. Truth Tables and Logical Equivalence Example 9: Construct a truth table for the compound proposition (p V q) Л r.

C. Truth Tables and Logical Equivalence Solution to example 9:

D. Implication and Equivalence If two propositions can be linked with “If.... then....”, then we have an implication. The implicative statement “if p then q” is written p => q and reads “p implies q”. p is called the antecedent and q is called the consequent. For example: Given p: Kato has a TV set, and q: Kato can watch TV, we have p => q: If Kato has a TV set, then Kato can watch TV.

C. Truth Tables and Logical Equivalence the truth table for p => q is:

C. Truth Tables and Logical Equivalence If two propositions are linked with “.... if and only if....”, then we have an equivalence. The equivalence “p if and only if q” is written p  q. p  q is logically equivalent to the conjunction of the implications p => q and q => p. Example: Consider p: I will pass the exam, and q: The exam is easy. We have p => q: If I pass the exam, then the exam is easy. q => p: If the exam is easy, then I will pass it. p  q: I will pass the exam if and only if the exam is easy.

C. Truth Tables and Logical Equivalence We can find the truth table for p, q by constructing the truth table of its logical equivalent (p => q) Л (q => p) So, the truth table for equivalence p  q is

E. Converse, Inverse, and Contrapositive The converse of the statement p => q is the statement q => p. Example: p => q: If it is a red car, then it’s a convertible q => p: If it’s a convertible, then it is a red car The truth table for converse is

E. Converse, Inverse, and Contrapositive Example 10: For p: the triangle is isosceles, and q: two angles of the triangle are equal, state p => q and its converse q => p.

E. Converse, Inverse, and Contrapositive Solution to example 10: p => q: If the triangle is isosceles, then two of its angles are equal. q => p: If two angles of the triangle are equal, then the triangle is isosceles

E. Converse, Inverse, and Contrapositive

the converse and inverse of an implication are logically equivalent. converse inverse

E. Converse, Inverse, and Contrapositive

p => q

E. Converse, Inverse, and Contrapositive For example, consider p: Sam is in the library and q: Sam is reading.

E. Converse, Inverse, and Contrapositive Example 11: Write down the contrapositive of: “All teachers drive blue cars”.

E. Converse, Inverse, and Contrapositive Solution to example 11:

F. Valid Arguments An argument is made up of a set of propositions, called the premise, that leads to a conclusion. An argument is usually indicated by a word such as ‘therefore’ or ‘hence’. Example: If George is at the beach, then he is getting sunburnt. George is at the beach. Therefore, George is getting sunburnt.

E. Converse, Inverse, and Contrapositive We set out arguments by separating the premise and the conclusion with a horizontal line. If George is at the beach, then he is getting sunburnt. George is at the beach. George is getting sunburnt. premise conclusion

E. Converse, Inverse, and Contrapositive So, from the propositions p => q and p, we are implying the conclusion q. We can write this argument in logical form as (p => q) Л p => q. To determine whether this argument is valid, we construct a truth table for this proposition, and see whether it is a tautology. We have a tautology, so our argument is valid. The conclusion we have made follows logically from the premise.

E. Converse, Inverse, and Contrapositive Example 12: Determine the validity of the following argument: If a triangle has three sides, then 2 +4 = = 7 Hence, a triangle has three sides.

E. Converse, Inverse, and Contrapositive Solution to example 12:

E. Converse, Inverse, and Contrapositive IMPORTANT: Proposition q is clearly false. However, this does not affect the validity of the argument. Logic is not concerned with whether the premise is true or false, but rather with what can be validly concluded from the premise. So if we started out with a wrong premise, our conclusion is wrong, but it is logically sound.

E. Converse, Inverse, and Contrapositive Example 13: Determine the validity of the following argument: If x is a natural number, then x is an integer. If x is an integer, then x is rational. Therefore, if x is a natural number, then x is rational.

E. Converse, Inverse, and Contrapositive Solution to example 13: