Use proof by contrapositive to prove the statement: For all real numbers, if m + n...

Use proof by contrapositive to prove the statement: For all real numbers, if m + n is irrational, then m or n is irrational.

Expert Solution

PROOF BY CONTRAPOSITIVE

In logical mathematics, contrapositive is a kind of conditional statement that is formed after negating both the terms. It reverses the direction of inference.
A simple statement is equal to its contrapositive statement.
If the simple statement is true them its contrapositive is also true or vice versa.

For example:-

Statement: If A then B (A----->B)

Contrapositive: If not A then not B (~A---->~B)

TRUTH TABLE OF CONTRAPOSITIVE

TO PROVE:- if m+n is an irrational number then m or n is irrational

Statement: if m+n is an irrational number then m or n is irrational

Contrapositive: if m+n is not irrational number then m or n is not irrational

i.e. if m+n is rational number then m or n is a rational number

m + n = p / q (since m+n is a rational no as per contrapositive statement)

m = p/q - n or n= p/q - m

since m+n was a rational number. Therefore any subtraction operation will also yield a rational number

m=rational number or n= rational number.

Hope this helps :)

venereology answered 1 year ago

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