1. MAT 3100 Introduction to Proof
Logic
&
Truth Tables

2. Logic Logic = the study of correct reasoning Use of logic
In mathematics:
to prove theorems
In computer science:
to prove that programs do what they are supposed to do

3. Propositions
A proposition is a statement or sentence that can be determined to be either true (T) or false (F).
Examples:
1+1=2
3+1=2
My dog ate my homework.
𝑛+1>3
4𝑥2+2𝑦−1

4. Operations on Truth Values
1. Negation (Not), ~p, ¬p,
Examples:
p: I am 21.
~p: I am not 21.
Rules
If p is T, then ~p is __ .
If p is F, then ~p is __.
Truth Table p ~p T F

5. Operations on Truth Values
2. Conjunction (And), p  q
Examples:
p: I am going to town.
q: It is going to rain.
p  q: I am going to town and it is going to rain.
Rules
p  q is T if both p and q are __.

6. Operations on Truth Values
2. Conjunction (And), p  q p q
p  q T F

7. Operations on Truth Values
3. (Inclusive) Disjunction (Or), p  q
Examples:
p: I am going to town.
q: It is going to rain.
p  q: I am going to town or it is going to rain.
Rules
p  q is T if p or q is __.

8. Operations on Truth Values
3. (Inclusive) Disjunction (Or), p  q p q
p  q T F

9. Operations on Truth Values
4. Exclusive Disjunction (Exclusive Or), p exor q
Rules
p exor q is T if p or q is __, but not both.

10. Operations on Truth Values
5. Implication (If…then…), p  q
Examples:
p: I am going to town. (hypothesis)
q: It is going to rain. (conclusion)
p  q: If I am going to town then it is going to rain.
Rules
p  q is F only when p is __ but q is __.

11. Operations on Truth Values
5. Implication (If…then…), p  q
Remark: Implication is defined different from normal (English) Language.
It is independent of ___________________
__________ between the hypothesis and conclusion.

12. Operations on Truth Values
5. Implication (If…then…), p  q p q
p  q T F

13. Operations on Truth Values
5. Implication (If…then…), p  q
If the hypothesis is F, then the implication is always T.
Examples:
p: I have $1 million in the bank.
q: I donate $10,000 to my church.
p  q:

14. Operations on Truth Values
Given p  q, we define the following related implications.
a. Converse of p  q : _______
b. Contrapositive of p  q: _______
c. Inverse of p  q: _______
We will see these related implications very often.

15. Operations on Truth Values
Examples:
If 𝑥=𝑦 then 𝑥2=𝑦2
Converse: If 𝑥2=𝑦2 then 𝑥=𝑦
Contrapositive: If 𝑥2≠𝑦2 then 𝑥≠𝑦
Inverse: If 𝑥≠𝑦 then 𝑥2≠𝑦2

16. Operations on Truth Values
Examples:
If 𝑥=5 then 𝑥+1=6
Contrapositive: If 𝑥+1≠6 then 𝑥≠5

17. Operations on Truth Values
Examples:
If 𝑥=1 then 𝑥2=1
Inverse: If 𝑥≠1 then 𝑥2≠1

18. Operations on Truth Values
6. Double Implication, Biconditional Proposition, (…if and only if…), p  q
Rules
p  q is T if p and q ___________________.

19. Operations on Truth Values
Summary p q
p  q
p  q
p  q
p  q T F

20. In Class Learning Activity
Classwork
Group Explorations