Propositions and Truth Tables

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

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

Negation of pLet 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 theNegation of a Proposition

p p

T FF T

p = The sky is blue.

p = It is not the case that the sky is blue.

p = The sky is not blue.

Conjunction of p and q: AND

Let p and q be propositions. The proposition “p and q,” denoted by pq 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 pq

T T T T F FF T FF F F

Disjunction of p and q: OR

Let p and q be propositions. The proposition “p or q,” denoted by pq, 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 pq

T T T T F TF T TF F F

Two types of OR

INCLUSIVE OR means “either or both”

EXCLUSIVE OR means “one or the other, but not both”

Two types of Disjunction of p and q: OR

INCLUSIVE OR means “either or both”

p q pq

T T T T F TF T TF F F

EXCLUSIVE OR means “one or the other, but not both”

p q pq

T T F T F TF T TF F F

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

Converse, Inverse, Contrapositive

Conditional p q

Contrapositive of p q is the proposition q p

.

Converse of p q is q p

Inverse of p q is p q

If you are not breathing, then you are not sleeping

p You are sleeping q you are breathing

If you are sleeping, then you are breathing.

If you are breathing, then you

are sleeping.

If you are not sleeping, then you are not breathing.

Find the conditional, converse, inverse and contrapositive:

Conditional p q

Contrapositive of p q is the proposition q p

Converse of p q is q p

Inverse of p q is p q

If the sun is shining, then it is warm outside.

p The sun is shinning q it is warm outside

If it is warm outside, then the sun is shining.

If the sun is not shining, then it is not warm outside.

If it is not warm outside,the sun is not shining.

Biconditional

Let p and q be propositions. The biconditional pq 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 pq.

p q pq

T T T T F FF T FF F T

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.

Logically Equivalent

• Compound propositions P and Q are logically equivalent if PQ 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: PQ (or by P Q)

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≡

(T/F) Conditional

(T/F) Converse

(T/F) Inverse

(T/F) Contrapositive

p mC = 100° q ABC is obtuse

If mC = 100°, then ABC is obtuse.

If ABC is obtuse then mC = 100°,

If mC ≠ 100°, then ABC is not obtuse.

If ABC is not obtuse, then mC ≠100°

(T/F) Conditional

(T/F) Converse

(T/F) Inverse

(T/F) Contrapositive

If ABC is isosceles, then it is equilateral

If ABC is not equilateral, then it is not isosceles

If ABC is not isosceles, then it is not equilateral

If ABC is equilateral, then it is isosceles

p ABC is equilateral q it is isosceles

(T/F) Conditional

(T/F) Converse

(T/F) Inverse

(T/F) Contrapositive

p G is the midpoint of KL q GQ bisects KL

If G is the midpoint of KL, then GQ bisects KL.

If GQ bisects KL, G is the midpoint of KL.

If G is not the midpoint of KL, then GQ does notbisects KL.

If GQ does not bisects KL then G is not the midpoint of KL.

Let R = “I work at this school” and D = “My name is Ms. D”

Translate the following symbols into sentences, and indicate T/F:

a. D R: ___ (T/F)

b. R D: ___ (T/F)

c. ~D ~R: ___ (T/F)

If my name is Ms. D, then I work at this school.

If I work at this school, then my name is Ms. D.

If my name is not Ms. D, then I do notwork at this school.

5. Given:a. If today is warm, the pool will be crowded.b. If it rains today, the pool will not be crowded.c. Either today is warm or I will wear a long-sleeved shirt.d. It will not rain today.Using W, P, R, S, & proper connectives (~, , etc.), express each sentence into symbolic form.

Let W represent “Today is warm.”Let P represent “The pool will be crowded.”Let R represent “It rains today.”Let S represent “I will wear a long-sleeved shirt.”

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a. w→ p

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b. r→˜ p

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c. w∨s

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d. ˜r

FTTF

FTTF

TTFT

TFFT

FTFT

TFTF

FFTT

FTFT

TFTT

TFFT

FFTF

FFTT

TFTF

TFTF

TFFT

TTFT

FTFT

TFFT

FTFT

TFFT

6. (p ^ w) ~(p ^ w) 7. p (w v) p (w v) 8. (v t) r (v t) r

9. (k p) g g (k p) 10. ~w r t (~w r) t 11. ~b (k p) (k p) ~b

12. (n j) e (n j) e 13. u (~h c) (~h c) u 14 (w ~ ~r) ~(w ~ ~ r)

15. f r ~t ( f r) ~t 16. (j u) ~y ~y (j u) (j u) ~y

TTFT

TTFT

TFTT

TFTT

FTTT

FFTF

TFFT

TTTF

FFFT

TTFT

FFTT

FFFF