Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce Haese and Haese Publications, 2004

Summary of Symbols SymbolReadMeaning pp “not p”Negation pqpq “p and q”Conjunction pqpq “p or q”Disjunction pqpq “p or q” Exclusive Disjunction p  q “if p then q”Implication pqpq “p if and only if q”Equivalence

Terms Tautology Contradiction Logical Equivalence Implication Converse Inverse Contrapositive

If two propositions can be linked with “If…, then…”, then we have an implication. p is the antecedent and q is the consequent SymbolReadMeaning pp “not p”Negation p  q “p and q”Conjunction p  q “p or q”Disjunction p  q “p or q”Exclusive disjunction p  q “if p then q”Implication Chapter 15E - Implication

Write down p  q in each of these cases: a) p: John is not at work. q: John is sick. a) p: My watch is slow. q: I will be late. b) p: The radio works. q: The power is on. Example 1

Truth Table for Implication p: It is Wednesday. q: I will have sushi for lunch. p  q: pq p  q TTT TFF FTT FFT If it is Wednesday, then I will have sushi for lunch.

Determine whether the statement p  q is logically true or false. If 5  4 = 20, then the Earth moves around the Sun. If the Sun goes around the Earth, then I am an alien. Example 2

Show that the following proposition is a tautology. Example 3

Implication and Venn Diagram p  q is same as P  Q P Q Let P = the set of the multiples of 9 Let Q = the set of the multiples of 3

Equivalence If two propositions are linked with “… if and only if…”, then it is an equivalence. SymbolReadMeaning pp “not p”Negation p  q “p and q”Conjunction p  q “p or q”Disjunction p  q “p or q”Exclusive disjunction p  q “if p then q”Implication p  q “p if and only if q”Equivalence

Write down p  q in each of these cases: a) p: We will play tennis. q: The weather is warm. a) p: Mary will pass math. q: The exam is easy. a) p: Madrid is in Spain. q: Spain is in Europe. Example 4 Note: p  q is the same as p  q and q  p

Truth Table for Equivalence pq p  q TTT TFF FTF FFT p : A polygon has 4 sides q : A polygon is a quadrilateral p  q : A polygon has 4 sides if it is a quadrilateral.

Equivalence and Venn Diagram P Q p  q is same as P = Q

Consider: p: My shoes are too small. q: My feet hurt. Implication: Converse: Inverse: Contrapositive: If my shoes are too small, then my feet hurt. If my feet hurt, then my shoes are too small. If my shoes are not too small, then my feet do not hurt. If my feet do not hurt, then my shoes are not too small. q  p q  pq  p  p   q p  q Section 15F - Converse, Inverse, and Contrapositive

Write the converse, inverse and contrapositive for: If a number is divisible by 10, then it ends in a zero. Example 5

Truth Table for Converse, Inverse, Contrapositive pq pp qqp  qq  p  p   q  q   p

Truth table for Converse, Inverse, Contrapositive pq pp qqp  qq  p  p   q  q   p TTFFTTTT TFFTFTTF FTTFTFFT FFTTTTTT What do you notice? Logical equivalence between 2 pairs of columns

 p   q  q   p p  q q  p CONVERSE INVERSE CONTRAPOSITIVE

Three propositions p, q and r are defined as follows: p: the water is cold. q: the water is boiling. r: the water is warm. a) Write one sentence, in words, for the following logic statement: (  p   q)  r b) Write the following sentence as a logic statement using symbols only. "The water is cold if and only if it is neither boiling nor warm" Example 6

Three propositions are defined as follows: p: The oven is working. q: The food supply is adequate. r: The visitors are hungry. a) Write one sentence, in words only, for each of the following logic statements. (i) q  r   p (ii)  r  (p  q) b) Write the sentence below using only the symbols p, q and logic connectives. "If the oven is working and the food supply is adequate then the oven is working or the food supply is adequate." Example 7

Homework 15E, pg 508 – #3, 4aceg, 5a, 7 15F, pg 509 – #1bd, 3ace