Tautology, contradiction and contingency

Hello friends, Welcome to my blog mathstips4u.

In my last post we have seen converse, Inverse and contrapositive of an implication and its examples. Some of the examples were left as exercise for you. That will be covered in this post. If you not still read that post, please read that post before watching this video.

In this post we are going to learn what is meant by tautology, contradiction and contingency and their examples.

Tautology (t):

A statement pattern which is true for all the combinations of the truth values of its component statements, is called a tautology. e.g. p v ~p

p	~p	p v ~p
T	F	T
F	T	T

 Contradiction(c):

A statement pattern which is false for all the combinations of the truth values of its component statements, is called a contradiction. e.g. p Ʌ ~p

p	~p	p Ʌ ~p
T	F	F
F	T	F

It is obvious that the negation of a tautology is a contradiction and vice versa.

Contingency:

A statement pattern which is neither a tautology nor a contradiction is called contingency.  e.g.  p v q

p	q	p v q
T	T	T
T	F	T
F	T	T
F	F	F

Ex. Determine whether the following statement pattern is a tautology or a contradiction or contingency.

1) (~p v q) → [p Ʌ (q v ~q)]

p	q	~p	~q	~p v q = a	q v ~q	p Ʌ (q v ~q) =b	a → b
T	T	F	F	T	T	T	T
T	F	F	T	T	T	T	T
F	T	T	F	T	T	F	F
F	F	T	T	T	T	F	F

The entries in the last column of the above truth table are neither all T nor all F.

Therefore, the given statement is neither a tautology nor a contradiction. It is a contingency.

Now we shall see exercise from my last post on converse, inverse and contrapositive.

Ex.  1) Write the converse, Inverse and contrapositive of an implication.

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

Solution:

Rewriting given statement by using if. …then.

If the women in the family are literate, then family become literate.

p: the women in the family are literate

q: a family become literate

its symbolic form is p → q

a)     Its converse is q → p:

i.e. If a family becomes literate, then the women in it are literate.

b) Its inverse is   ̴p →   ̴q

i.e. If the women in a family are not literate, then family does not become literate.

c) Its contrapositive is ~q →~p

i.e. If a family does not become literate, then the women in it are not literate.

 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 same meaning.

Solution:

Let p: a person is social and q: a person is happy

The above statement patterns are written in symbolic form.

a) p → q          Implication

b) ~p → ~q     Inverse

c) ~q → ~p     contrapositive

d) q → p         converse

We know that implication and its contrapositive are same.

Therefore, (a) and (c) are same.

And converse and inverse of an implication are same.

Therefore, (b) and (d) are same.

Ex. 3. Ex. Rewrite the following statements without using the conditional form:

2) If it is cold we wear woolen clothes.

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

Solution:

2) p: It is cold q: we wear woolen clothes

Its symbolic form is p → q.

We know that p → q≡~p v q.

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

i.e. It is not cold or we wear woolen clothes.

3) rewriting above statement as

If I take cold water bath, then I can catch cold.

P: I take cold water bath q: I can catch cold.

Its symbolic form is p → q.

        And as above p → q≡~p v q.

Then the statement without conditional is

‘I do not take cold water bath or I can catch cold’.

In this way we have seen tautology, contradiction and contingency and their examples.

My next post we are going to learn what is meant by quantifiers and quantified statements.

Thanking you.