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Implications

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Implications - Lesson Summary

Implication: A compound statement formed using the connecting words "If - then..." is called an implication or a conditional statement.

Now, the compound statement can be expressed as "if p, then q".

If statement p is true, then statement q will be true.

Such compound statements are called implications or conditional statements.

It is formed using the connecting words "if-then."

p implies q is represented as p ⇒ q.

A compound statement, "if p, then q," is always true, except in the case when p is true and q is false, because a true statement cannot imply a false statement.

Consider the statement 'If 3 is a factor of 9, then it is also a factor of 18.'

The component statements are:

p: 3 is a factor of 9.

q: 3 is a factor of 18.

Statement p ⇒ q : 3 is a factor of 9 implies that 3 is also a factor of 18.

p ⇒ q can also be expressed as:

• If p, then q.

• ~q  if ~p.

• p only if q.

• p is necessary condition for q.

• q is sufficient condition for p.

Converse, inverse and contrapositive of an implication


Consider an implication, p ⇒ q

The converse of p ⇒ q is q ⇒ p.

The inverse of p ⇒ q is ~p ⇒ ~q

The contrapositive of p ⇒ q is ~q ⇒ ~p.

Examples:

1) Converse, inverse and contrapositive of the implication — if two sides of a triangle are equal, then it is an isosceles triangle.


The component statements are:

p: Two sides of a triangle are equal.

q: The triangle is isosceles.

The implication of the statements is p ⇒ q.

The conditional statement,

p ⇒ q: If two sides of a triangle are equal, then it is an isosceles triangle.

Converse:

q ⇒ p: If a triangle is isosceles, then two of its sides are equal.

Inverse:

~p ⇒ ~q: If two sides of a triangle are not equal, then it is not an isosceles triangle.

Contrapositive:

~q ⇒ ~p: If a triangle is not isosceles, then two of its sides are not equal.

2) Implication: If the sides of a triangle are equal, then its angles are also equal.

Converse: If the angles of a triangle are equal, then its sides are also equal.

Here, both the statements are true.

Such statements can also be expressed as — the sides of a triangle are equal if and only if its angles are equal. Such compound statements are called a bi-implication or bi-conditional.

Bi-implication: A compound statement involving statements p and q, which is in the form of "p if and only if q," is called a bi-implication or bi-conditional.

p if and only if q is represented by p ⇒ q.

p if and only if q or p ⇒ q can also be expressed as:

• q if and only if p

• p iff q

• p is a necessary and sufficient condition for, q and vice-versa

If p and q have the same truth values (either true or false), then the bi-implication is true.

If either p or q is false, then the bi-implication is false.

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