00S1501/10113201 Question 15 Which one of the allernatives is a proof by contrap
ID: 3710192 • Letter: 0
Question
00S1501/10113201 Question 15 Which one of the allernatives is a proof by contrapositive of the statemerk If x-x+4 is not divisible by 4 then x even." 1. Required to prove: If x-x+ 4 is not divisbla by 4 then x even. Proot Suppose x is odd. Lat x -2k 1, than we have to prove that x) - x + 4 is dvislble by 4 x+4 (1-12k+1)+4 - (2k 1)(4k+4k +1)-2k-1+4 gk 12k2 + 4k + 4 4(2+3+ k + 1), which is divisible by 4. (4 mutipilied by any integer is divieible by 4) 2. Required to prove. If x3-x + 4 is not disible by 4 ten x even. Proof Assume that x-x4 is not divisible by 4. Then x can be even or odd. We sssume that x is odd Letx- 2k+ 1, then x-x4 (2k 1)-(2x+ 1) 4 (2k+ 1)(4k2 4k +1)-2k-1+4 Bk+8 2k+4k+4k +1-2k-1+4 s 8k+ 12k2+ 4k +4 4(2k+3k2+ k+ 1), which is divisible by 4. (4 multiplied by any integer is divisible by 4) But this is a contradiction to our original assumption. Therefore x must be even if x-x+ 4 is not divisible by 4, 3. Required to prove: lf x,-x + 4 is not divisible by 4 then x even. Proof Let x4 be an even element of Z. We can replace x with 4 in the expressionx -x+4 x3-x+4 =64-4+4 = 64 which is divisible by 4. Required to prove: If x. x + 4 is not divisible by 4 then x even. Proof 4. Assume that x is even, ie x then x-x +4 4k (4k),-(40 + 4 64k3-4k + 4 4(16k3-k + 1), which is divisible by 4. 31Explanation / 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.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.