Converse, Inverse and contrapositive of an implication

Hello friends, Welcome to my blog mathstips4u.

In my last post we have seen logical equivalence and some of their examples.

In this post we are going to learn converse, inverse and contrapositive of an implication and some of their examples.

If p → q is an implication, then there arises following three implications

1.     q → p is called converse of p → q

2.      ~p →   ~q is called inverse of p → q

3.       ~q →   ~p is called contrapositive of p → q

We shall see the truth table of converse, inverse and contrapositive of an implication.

p	q	~p	~q	p → q conditional	q → p converse	~p→~q inverse	~q → ~p contrapositive
T	T	F	F	T	T	T	T
T	F	F	T	F	T	T	F
F	T	T	F	T	F	F	T
F	F	T	T	T	T	T	T
1	2	3	4	5	6	7	8

Observer that column no. (5) and column no. (8) are identical therefore

P →q ≡~q →~p i.e. conditional and its contrapositive are same.

and column no. (6) and column no. (7) are identical therefore

    q →p ≡~p → ~q i.e. converse and inverse of an implication are same

Ex.  1) Write the converse, Inverse and contrapositive of the following implications.

i)  If a sequence is bounded, then it is convergent.

ii) A family becomes literate if the women in it are literate.

Solution:

i) Let p: A sequence is bounded. q: It is convergent.

a) Its converse is q → p:

i.e. If a sequence is convergent then it is bounded.

b) Its inverse is   ~p → ~q

i.e. If a sequence is not bounded, then it is not convergent.

c) Its contrapositive is   ~q → ~p

i.e. If a sequence is not convergent then it is not bounded.

The second example is left as an exercise for you. Try it or wait till next post uploaded in my blog  “mathstips4u”.

Ex. 2) Consider following statements.

a)     If a person is social, then he is happy.

b)     If a person is not social, then he is not happy.

c)     If a person is unhappy, then he is not social.

d)     If a person is happy, then he is social.

Identify the pairs of statements having the same meaning.

This is also left as an exercise for you. Try it or wait till next post uploaded in my blog “mathstips4u”.

Ex. Rewrite the following statements without using the conditional form:

1) If prices increase then the wages rise.

2) If it is cold we wear woolen clothes.

3)I can catch cold if I take cold water bath.

Solution:  1) a: prices increase.    q: wages rise.

Given statement written in symbolic form as p → q

In my last post we have seen p → q≡~p v q.

Therefore ~p v q is the statement without the conditional form.

i.e.  Prices do not increase or the wages rise.

The remaining examples are left as an exercise for you. Try it or wait till next post uploaded in my blog “mathstips4u”.

In this way we have seen how to write converse, inverse and contrapositive of an implication.

My next post is on tautology, contradiction and contingency.