Converse Inverse and Contrapositive
- 5 years ago
Converse, Inverse and contrapositive of an implication.
Hello friends, Welcome to my channel mathstips4u.
In my last video we have seen logical equivalence and some of their examples.
In this video 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 video uploaded in my channel “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 video uploaded in my channel “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 video 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 video uploaded in my channel “mathstips4u”.
In this way we have seen how to write converse, inverse and contrapositive of an implication.
My next video is on tautology, contradiction and contingency.
If you like my video, please subscribe my channel, like it, share it and comment it.
Thanking you for watching my video.
Hello friends, Welcome to my channel mathstips4u.
In my last video we have seen logical equivalence and some of their examples.
In this video 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 video uploaded in my channel “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 video uploaded in my channel “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 video 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 video uploaded in my channel “mathstips4u”.
In this way we have seen how to write converse, inverse and contrapositive of an implication.
My next video is on tautology, contradiction and contingency.
If you like my video, please subscribe my channel, like it, share it and comment it.
Thanking you for watching my video.