Discrete Structure / Discrete Mathematics Lecture 2: CONDITIONAL and BICONDITIONAL PROPOSITIONS

CONDITIONAL CONNECTIVE

Let p and q be propositions. The implication p q is the proposition that is false only when p is true and q is false; otherwise it is true. p is called the hypothesis and q is called the conclusion. The connective is called the conditional connective

Example 2.1

Construct the truth table of the implication p q

Solution.The truth table is:

.Example 2.2

Show that p q p q.

Solution.The truth table is:

pqp p qp q

FFTTT

FTTTT

TFFFF

TTFTT

It follows from the previous example that the proposition pq is always true if the hypothesis p is false, regardless of the truth value of q.

In terms of words the proposition p q also reads:

(a) if p then q.(b) p implies q(c) p is a sufficient condition for q(d) q is a necessary condition for p(e) p only if q

Example 2.3

Use the if-then form to rewrite the statement: I am on time for work if I catch the 8:05 bus."

Solution.If I catch the 8:05 bus then I am on time for work.

In propositional functions that involve the connectives and the order of operations is that is performed first and is performed last.

Example 2.4

a. Show that (p q) p q.b. Find the negation of the statement: If my car is in the repair shop, then I cannot go to class.

Solution.c. Show that (p q) p q.pqp q(p q) qp q

FFTFTF

FTTFFF

TFFTTT

TTTFFF

a. My car is in the repair shop and I can get to class."

CONVERSE, OPPOSITE and CONTRAPOSITIVE

The converse of p q is the proposition q p. The opposite or inverse of p q is the proposition p q. The contrapositive of p q is the proposition q p.

Example 2.5

Find the converse, opposite, and the contrapositive of the implication:

If today is Thursday, then I have a test today."

Solution.

The converse If I have a test today then today is Thursday.

The opposite If today is not Thursday then I don't have a test today.

The contrapositive If I don't have a test today then today is not Thursday

Example 2.6

Show that p q q p.

pq q p q pp q

FFTTTT

FTFTTT

TFTFFF

TTFFTT

Example 2.7

Using truth tables show the following:a. p q q pb. p q p q

Solution:

a. p q q p //example 2.2: p q p q //therefore: q p q p //and: p q q p

pq p q p q q p

FFTTTT

FTTFTF

TFFTFT

TTFFTT

pqp qqpq p

FFTFFT

FTTTFF

TFFFTT

TTTTTT

Solution:

b. p q p q

pqp qp qp q

FFTTTT

FTTFFT

TFFTTF

TTFFTT

BICONDITIONAL CONNECTIVE

The biconditional proposition of p and q, denoted by p q, is the propositional function that is true when both p and q have the same truth values and false if p and q have opposite truth values. Also reads, p if and only if q" or p is a necessary and sufficient condition for q."

Example 2.8

Construct the truth table for p q:

Solution:

Example 2.9

Show that the biconditional proposition of p and q is logically equivalent to the conjunction of the conditional propositions p q and q p:

Solution:

pqpqqpq p(p q)( q p)

FFTFFTT

FTTTFFF

TFFFTTF

TTTTTTT

Lecture 2: Conditional and Bi-Conditional Propositions Page | 2