Which one of the alternatives is a proof by contrapositive of the statement If x³ -x + 4 is not divisible by 4, then x even. O a. 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 x³ -x + 4 is divisible by 4. x³-x+4= (2k + 1)³-(2k + 1) + 4 = (2k + 1) (4k² + 4k+1)-2k-1+4 = 8k³ + 8k² + 2k + 4k² + 4k +1-2k-1+4 = 8k³ + 12k² + 4k+ 4 = 4(2k³ + 3k² + k + 1), which is divisible by 4. (4 multiplied by any integer is divisible by 4) O b. Required to prove: If x3 -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 is odd. Let x = 2k + 1, then x³ -x +4 = (2k+1)³-(2k + 1) +4 = (2k + 1)(4k² + 4k+1)-2k-1+4 = 8k³ + 8k² + 2k + 4k² + 4k +1-2k-1+4 = 8k³ +12k² + 4k +4 = 4(2k³ + 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 x³ -x + 4 is not divisible by 4. O c. Required to prove: If x³ -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 x³-x +4. x³ -x +4= 64-4+4= 64 which is divisible by 4. O d. Required to prove: If x³ -x + 4 is not divisible by 4, then x even. Proof: Assume that x is even, i.e. x = 4k, then x³ -x + 4 = (4k)³ - (4k) + 4 = 64k³ - 4k + 4 = 4(16k³k+1), which is divisible by 4.
Which one of the alternatives is a proof by contrapositive of the statement If x³ -x + 4 is not divisible by 4, then x even. O a. 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 x³ -x + 4 is divisible by 4. x³-x+4= (2k + 1)³-(2k + 1) + 4 = (2k + 1) (4k² + 4k+1)-2k-1+4 = 8k³ + 8k² + 2k + 4k² + 4k +1-2k-1+4 = 8k³ + 12k² + 4k+ 4 = 4(2k³ + 3k² + k + 1), which is divisible by 4. (4 multiplied by any integer is divisible by 4) O b. Required to prove: If x3 -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 is odd. Let x = 2k + 1, then x³ -x +4 = (2k+1)³-(2k + 1) +4 = (2k + 1)(4k² + 4k+1)-2k-1+4 = 8k³ + 8k² + 2k + 4k² + 4k +1-2k-1+4 = 8k³ +12k² + 4k +4 = 4(2k³ + 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 x³ -x + 4 is not divisible by 4. O c. Required to prove: If x³ -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 x³-x +4. x³ -x +4= 64-4+4= 64 which is divisible by 4. O d. Required to prove: If x³ -x + 4 is not divisible by 4, then x even. Proof: Assume that x is even, i.e. x = 4k, then x³ -x + 4 = (4k)³ - (4k) + 4 = 64k³ - 4k + 4 = 4(16k³k+1), which is divisible by 4.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 50E: Show that if the statement 1+2+3+...+n=n(n+1)2+2 is assumed to be true for n=k, the same equation...
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Transcribed Image Text:Which one of the alternatives is a proof by contrapositive of the statement If x³ -x + 4 is not divisible by 4, then x even.
a. 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 x³ -x + 4 is divisible by 4.
x³ -x + 4 = (2k + 1)³-(2k + 1) + 4
= (2k + 1)(4k² + 4k +1) - 2k-1+4
= 8k³ + 8k² + 2k + 4k² + 4k +1-2k-1+4
= 8k³ + 12k² + 4k + 4 = 4(2k³ + 3k² + k + 1), which is divisible by 4. (4 multiplied by any integer is divisible by 4)
b. Required to prove: If x³ -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 is odd.
Let x = 2k + 1, then x³ -x +4
= (2k+1)³(2k + 1) +4
= (2k + 1) (4k² + 4k+1)-2k-1+4
= 8k³ + 8k² + 2k + 4k² + 4k +1-2k-1+4
= 8k³ + 12k² + 4k + 4
= 4(2k³ + 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 x³ -x + 4 is not divisible by 4.
c. Required to prove: If x³ -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 x³ - x + 4.
x³ -x + 4 = 64-4 + 4 = 64 which is divisible by 4.
O d. Required to prove: If x³ -x + 4 is not divisible by 4, then x even.
Proof: Assume that x is even, i.e. x = 4k, then
x³ -x + 4 = (4k) - (4k) + 4 = 64k³ - 4k + 4 = 4(16k³ − k + 1), which is divisible by 4.
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