i to be having trouble proving ns statemenns listed below using proof through Contrapositive. I have negate the statements as required and climate i prove that $n$ ins odd if and only if $7n+4$ is odd. However, native there ns am stuck and am no sure how come prove ns first component the if $n$ is negative integer, climate $n$ is odd. Mine work-related ins show below.

You are watching: Prove that if n is a positive integer, then n is even if and only if 7n + 4 is even.

Initial Statement: if n is a Confident integer climate n ins even if and only if is 7n+4 is even.Contrapositive: If n is negati have creature then n is odd if and also only if 7n+4 ins odd.As such by interpretation of odd: n = 2k+1instead of n:=7(2k+1)+4=14k+7+4=14k+11=2(7k)+11Therefore, n ins strange and 7n+4 ins odd.thats as far as i obtained and also ns dont also know if what ns go above ins also appropriate though. Thanks.


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edited jan 29 "15 in ~ 23:10

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asked january 29 "15 at 23:05
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Tbelow yes, really ins no should prove through contraConfident (unless your teacshe ins requirinns friend come carry out the for some reason). The adhering to direct proof functions just fine.

Problem: intend $ninstartupcuba.orgbbZ$ ($n$ being Hopeful ins yes, really not necessary). Climate $n$ is also if and only $7n+4$ is even.

Proof.

($ o$): mean $n$ is even. Then $n=2ell$, where $ellinstartupcuba.orgbbZ$. Thus, we have that$$7n+4=7(2ell)+4=14ell+4=2(7ell+2)=2m,$$wright here $m=7ell+2$ and also $minstartupcuba.orgbbZ$. Thus, the front directi~ above ins true.

($leftarrow$): suppose $7n+4$ is even. Think about wcap happens once $n$ ins odd or even.

$n$ odd: we have actually $n=2ell+1$ because that part $ellinstartupcuba.orgbbZ$ and so $$7(2ell+1)+4=14ell+11=2(7ell+5)+1=2m+1,$$ wbelow $m=7ell+5$ and $minstartupcuba.orgbbZ$.$n$ even: we have actually $n=2ell$ because that part $ellinstartupcuba.orgbbZ$ and so$$7(2ell)+4=14ell+4=2(7ell+2)=2m,$$wbelow $m=7ell+2$ and $minstartupcuba.orgbbZ$.

the over analysis mirrors that $n$ is also if and also just if $7n+4$ is even, together desired.


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answer january 29 "15 in ~ 23:23
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i thsquid girlfriend want to reexpression the Initial statement as:

"Lens $n$ be a Hopeful integer. Then $n$ is also if and also just if $7n+4$ is even."

with "if and also just if" proofs, we would certainly speak to the front direction of this evidence

"If $n$ ins also then $7n+4$ ins even"

This ins identical to its contrapositive, "If $7n+4$ ins not even then $n$ ins not even." together you sharp out, thins contraHopeful is ns same together speak "If $7n+4$ ins odd then $n$ ins odd"

Next, ns backward directitop top ins

"If $7n+4$ ins also climate $n$ ins even"

i beg your pardon is indistinguishable to ns contraConfident statemenns "If $n$ ins odd climate $7n+4$ is odd."

therefore come finish her proof, girlfriend have to complete the evidence in ns front directi~ above (whether friend perform it straight or prove the contrapositive) and the same, similar thing friend have to show the behind direction. When girlfriend have actually da this, girlfriend might break up the $n$ ins also iff $7n+4$ is even.


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answered january 29 "15 in ~ 23:27

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her statement of ns contraOptimistic ins correct, except the ns first "negative" need to it is in "positive".

the dispute you have actually offered proves effectively the if $n$ ins odd, climate $7n+4$ ins odd.

It continues to be to prove the if $7n+4$ is odd, then $n$ is odd. Right here ins a feasible proof.

Lens $n$ it is in an essence and also i think that $7n+4$ ins odd. Currently $6n+4$ ins obvioucunning even, and also the difference between an odd and an even number is odd, Because of this $(7n+4)-(6n+4)$ ins odd. That is, $n$ ins odd. This completens ns proof.


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answer jan 29 "15 at 23:32

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