Which one of the altematives is a proof by contrapositive of the statement If x3

Question

Which one of the altematives is a proof by contrapositive of the statement If x3-x+4 is not divisible by 4 then x even. 1. Required to prove. If x3-X + 4 is not divisible by 4 then x even. Proof Suppose x is odd. Let x " 2k + 1,then we have to prove that x, x-x+4 (2k 1)- (2k +1)+4 -(2k+1X4k+ 4k +1)-2k-1+4 - 8k+ 8k+2k+4k+ 4k +1 -2k-1+4 -x+ 4 is divisible by 4. 8k3 + 12k2 + 4k + 4 2+3k+ k+ 1), which is divisible by 4. (4 multiplied by any integer is divisible by 4) 2. Required to prove: Ifx3-x+ 4 is not divisible by 4 then x even. Proof Assume that x3-x + 4 is not divisible by 4. Then x can be even or odd we assume that x i odd Let x 2k+ 1, then x3-x+4-(2k+1P-(2k+ 1)+4 - (2k + 1 4k2+4k 1)-2k-1+4 - 8k+ 8k2 + 2k +4k2 4k +1-2k-1+4 8k3 + 12k2 + 4k + 4 4(2k+3k2 k+ 1), which is divisible by 4. (4 multlplied by any integer is divisible by 4) But this is a contradiction to our original assumption. Therefore x must be even ifx3-x+4 is not divisible by 4. 3. Required to prove: Ifx3-x+4 is not divisible by 4 then x even. Proof Let x 4 be an even element of Z. We can replace x with 4 in the expression x-x·4 (4-(4) 4 64-4+ 4 64 which is divisible by 4. Required to prove: lf x3-x + 4 is not divisble by 4 then x even. Proof Assume that x is even, ie x # 4k, then x - x+4 = (4k)-(4k) + 4 -64k-4k +4 4(16k-k + 1), which is divisible by 4 4.

Explanation / Answer

Option 2 is correct choice.

proof by contradiction :

In this approach we can Prove the statement is valid by taking opposite meaning of actual statement here the result fails so we can say our assumption is wrong.

Since opposite meaning of actual statement fails indicates that original statement is true

The whole approach refers to proof by contradiction.

From the question

X is even, In proof by contradiction assume that x is an odd value.

Here the result is always opposite meaning of original statement.

so the result here we got is expression is divisible by 4 it is opposite meaning of original statement.

So we can justify that x must be an even value.

Please rate my answer.

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