Conditional and Biconditional Statements

Logical Equivalence

Two propositions (simple or compound) are said to be logically equivalent if they have identical truth values.

If p is logically equivalent to q, we denote p q.

We have already constructed the truth table of (~p)q in earlier class. Recall the same truth table.

Truth Table for (~p)q

Note that the last column of above table and last column of earlier table are identical. This implies p q is logically equivalent to (~p)q.

C5

We explain C5 with the following example.

Consider the statement "either the monsoon break in June or the rivers are dried"

Let p: The monsoon does not break in june

q: rivers are dried

Clearly 'if p then q' is true if p is true and q is true. That is, the conditional statement of the above statement is "If the monsoon does not break in June then rivers are dried".

This statement is equivalent to "The monsoon breaks in the month of June or the rivers are dried" which is symbolically written as (~p)q.

We know from C5 that

Taking negation on both sides, we have

Using Demorgan's law to L.H.S, we have

Thus we define

Negation of Conditional Statement

C6

Example:

Write the negation of the conditional statement

"If the weather is cold then it will snow"

The negation of the given statement is "The weather is cold and it will not snow"

Contrapositive of a Conditional Statement

We know that

C7

The conditional statement is equivalent to it contrapositive or

Example:

Consider the conditional statement "If a quadrilateral is a square, then all the sides are equal".

The contrapositive of this statement is "If all the sides of a quadrilateral are not equal, then it is not a square".

The Biconditional statement

If two simple statements p and q are connected by the connective 'if and only if', then the resulting compound statement is called the biconditional statement. Symbolically it is represented by p q.

Example: An integer is even if and only if it is divisible by 2. It is a biconditional having the truth value T.

The biconditional statement p q is true when either both p and q are true or both p and q are false.

The truth table for biconditional is given below

For further reference, we write the truth values of biconditional statements as follows.

B1:

pq has the truth value T if both p and q have the same truth value.

That is

p is true and q is true p q is true.

p is false and q is false p q is true.

B2

pq has truth value F if p and q has opposite truth values.

That is

If p has truth value T and q has the truth value F, pq has truth value F.

Also,

if p has the truth value F and q has the truth value T, p q has truth value F.

Another way of defining biconditional statement is given below.

B3

If p and q are propositions, then the conjunction of conditionals p q and q p is called a biconditional proposition.

Let us construct the truth table of (p q) (q p)

The last column of above table is identical to the last column of earlier table.