: you are trying to Prove the tallowiung Proposition by using contrapositive pro

Question

: you are trying to Prove the tallowiung Proposition by using contrapositive proof n in Proposition 1 : If nsa poesitive integer sinch that n 2 mod 4 or n 3 mod 4, then not a perfect square Which of the followings should be the first sentence of your proof (A) Suppose n is a positive integer such that n 2 lnod 4 or n 3 mod 4 (B) Suppose n is a postive iteger such that n - 2 mod 4 and n 3 mod 4 (C) Suppose n is not a perfect squark (D) Suppose n is a perfect asquare. Definition: A real valued function ( r) is called one-to-one ifri22 for any real numbers ri and r2, then /(ri)f(x2). Suppose you are trying to prove that r = a s l is one-to-one by using contrapositive proof: Which of the followings should be the first sentence of your proof? (A) Assume ri (B) Assume an- (C) Assume/(r.)/(ra) for so me ri, r2 (D) Assume f(r.) = /(r2) for some ri'#2

Explanation / Answer

Contrapositive:Switching the hypothesis and conclusion of a conditional statement and negating both.

For example, the contrapositive of "If it is raining then the grass is wet" is "If the grass is not wet then it is not raining."

3)Hypothesis is "n is a positive integer such that n = 2 mod 4 or n = 3 mod 4"

Conclusion is "n is not a perfect square"

In order to get the contrapositive we should switch the hypothesis and conclusion of a statement and negate both.

so contrapositive of given statement is "if n is a perfect square then n is positive integer such that n not equal to 2 mod 4 and n not equal to 3 mod 4".

so,first condition we have to consider is "n is a perfect square"   Option D is the correct answer

4)

Hypothesis is "x1 not equal to x2 for any real number x1,x2"

Conclusion is "f(x1) is not equal to f(x2)"

In order to get the contrapositive we should switch the hypothesis and conclusion of a statement and negate both.

so contrapositive of given statement is "if f(x1)=f(x2) for any real numbers x1 and x2 then x1=x2 ".

so,first condition we have to consider is "f(x1)=f(x2) for any real numbers x1 and x2" Option D is the correct answer

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