1. Propositions and Truth Tables

2. Proposition: Makes a claim that may be either true or false; it must have the structure of a complete sentence.

3. Are these propositions? Over the mountain and through the woods.
All apples are fruit.
The quick, brown fox.
Are you here?
2 + 3 = 23 NO
YES NO NO
YES

4. Negation of p
Let p be a proposition. The statement “It is not the case that p” is also a proposition, called the “negation of p” or p (read “not p”)
Table 1.
The Truth Table for the
Negation of a Proposition
p p
T F
F T
p = The sky is blue.
p = It is not the case that the sky is blue.
p = The sky is not blue.

5. Conjunction of p and q: AND
Let p and q be propositions. The proposition “p and q,” denoted by pq is true when both p and q are true and is false otherwise. This is called the conjunction of p and q.
Table 2. The Truth Table for the Conjunction of two propositions
p q pq
T T T
T F F
F T F
F F F

6. Disjunction of p and q: OR
Let p and q be propositions. The proposition “p or q,” denoted by pq, is the proposition that is false when p and q are both false and true otherwise.
Table 3. The Truth Table for the Disjunction of two propositions
p q pq
T T T
T F T
F T T
F F F

7. Two types of OR INCLUSIVE OR means “either or both”
EXCLUSIVE OR means “one or the other, but not both”

8. Two types of Disjunction of p and q: OR
INCLUSIVE OR means “either or both”
p q pq
T T T
T F T
F T T
F F F
EXCLUSIVE OR means “one or the other, but not both”
p q pq
T T F
T F T
F T T
F F F

9. Implications If p, then q p implies q if p, q p only if q
p is sufficient for q
q if p
q whenever p
q is necessary for p
Proposition p = antecedent
Proposition q = consequent

10. Converse, Inverse, Contrapositive
Conditional p  q If you are sleeping, then you are breathing.
Converse of p  q is q  p If you are breathing, then you are sleeping.
Inverse of p  q is p  q If you are not sleeping, then you are not breathing.
Contrapositive of p  q is the proposition  q  p If you are not breathing, then you are not sleeping.

11. Find the converse, inverse and contrapositive:
Conditional p  q If the sun is shining, then it is warm outside.
Converse of p  q is q  p
Inverse of p  q is p  q
Contrapositive of p  q is the proposition  q  p

12. Biconditional
Let p and q be propositions. The biconditional pq is the proposition that is true when p and q have the same truth values and is false otherwise. “p if and only if q, p is necessary and sufficient for q”
Table 6. The Truth Table for the biconditional pq.
p q pq
T T T
T F F
F T F
F F T

13. Logical Equivalence
An important technique in proofs is to replace a statement with another statement that is “logically equivalent.”
Tautology: compound proposition that is always true regardless of the truth values of the propositions in it.
Contradiction: Compound proposition that is always false regardless of the truth values of the propositions in it.

14. Logically Equivalent
Compound propositions P and Q are logically equivalent if PQ is a tautology. In other words, P and Q have the same truth values for all combinations of truth values of simple propositions.
This is denoted: PQ (or by P Q)