Mathematical Reasoning, Contrapositive Method, Validity of Statements by Contradiction

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Contrapositive Method:

Let be not true. Then is not true

is an even integer

Either is even or is even or both and are even

is not true

Thus is false

is false

Hence “If -then ” is a true statement.

Validity of Statements by Contradiction:

Here to check whether a statement is true, we assume that is not true i.e. is true.

Then we arrive at some result which contradicts our assumption. Therefore, we conclude that is true.

Example:

Verify by the method of contradiction is irrational.

Solution:

Let be the statement given by is irrational.

We assume that is rational

, where and are integers having no common factor.

Divides

Divides

For some integer

Divides

Divides

Thus, is common factor of both and. This contradicts that and have no common factor.

So, our assumption is rational is wrong. Hence the statement “ is irrational”, is true.

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