Question

Prove that if n is an integer then 3n + 2 is even, then n is even

Answer 1

we have to prove two things? : 1) 3n+2 is even , 2) n is even

Answer 2

no, it's prove n is even GIVEN 3n + 2 is even.

Answer 3

ok , thanks

Answer 4

If 3n + 2 is even, then 3n is even

Answer 5

if 3n is even then n is even.

Answer 6

Given : 3n+2 is even
To prove : n is even

3n+2 = 2k

3n = 2k-2

3n = 2(k-1)

Answer 7

not true for n=1 ?
3+2=5<---not even

Answer 8

@hartnn GIVEN that 3+2 is even, n is even

Answer 9

IF 3n+2 is even , prove that is even. @hartnn

Answer 10

But it's not so no guarentee is made.

Answer 11

lets clear it from @lgbasallote what the exact question is...

Answer 12

@lgbasallote ?

Answer 13

Just out of interest whats the general way to prove something is even, divide by 2?

Answer 14

yes

Answer 15

if we prove a number is of form 2k that proves it is even

Answer 16

right I see

Answer 17

btw any1 here mind taking a look at my question?
http://openstudy.com/updates/5078b9d7e4b02f109be44f95

Answer 18

sorry i just came back....anyway...i'd llike to see how proving by contradiction is done.. direct proof is too easy

Answer 19

but whats the question ?

Answer 20

the statement in the blue box

Answer 21

forgot to mention by the way....that the direct proof done here was wrong

Answer 22

they took n as the condition... 3n + 2 is supposed to be the condition

Answer 23

so 3n + 2 = (a different n)?

Answer 24

if 3n+2 is even , n is even.
thats the question and u need to prove that using contradiction, right ?

Answer 25

if you use direct proof...

it should be 3n + 2 = 2x

then prove n is even

Answer 26

but like i said...should be contradiction though

Answer 27

well okay thats easy enough, too

Answer 28

hmmm then let's see you try

Answer 29

All you have to say is that assume that given 3n + 2 is even, n is not even
Then n has to be odd. Thus, 3n + 2 can be represented as 2k + 1 for some k

Answer 30

Given : 3n+2 is even
To prove : n is even

to prove by contradiction, lets assume the opposite -
lets assume n is odd,

3n+2 = 2k

3n = 2k-2

3n = 2(k-1)

so we got the right side as even number,

but we assumed n is odd, so left 3n becomes odd - contradiction

Answer 31

@sara12345 you cannot say "assume that n is odd" and then equate it to 2k

Answer 32

@sara12345 how is that contradiction

Answer 33

we do that always in proof by contradiction

Answer 34

RHS = even , LHS = odd => contradiction

Answer 35

i was referring to your solution

Answer 36

How can you prove that it equals 2k, though?

Answer 37

if 3n + 2 is odd, then it can be expressed 3n + 2 = 2k + 1. solving for n gives \[n = \frac{2k}{3} - 1\] substituting back for n gives
\[3 (\times \frac{2k}{3} - 1) + 2 = 2k + 1so\]
so \[2k - 1 = 2k + 1 \]

which is absurd

Answer 38

hmm seems legit

Answer 39

so in proof by contradiction...you still substitute back huh

Answer 40

As @JamesWolf showed for you,

Answer 41

you take the contradiction, show that it is not internally consistent, and therefore it cannot be true.

Answer 42

no its not legit

Answer 43

ive been an idiot int he first step

Answer 44

@JamesWolf no, you inadvertently put up my proof.

Answer 45

in proof by contradiction, we assum e the opposite of what we need to prove as true, and proceed, not the opposite of given conditions

Answer 46

if you assume n is odd and 3n + 2 is even...

3(2k + 1) + 2

6k + 3 + 2

6k + 2 + 3

2(3k + 1) + 3

so even + odd would be odd...so contradiction

i suppose that works as well

Answer 47

@sara12345 yes you're right, sorry.

Answer 48

yes its right actually i managed to not divide by 3 at the start, but luckily i messed up by not multiplying - 1by 3 at the end

Answer 49

oh so you take the negation of q then proceed from there?

Answer 50

anyway...is my proof right?

Answer 51

lgba ur proof is more correct as it shows the assumption n is odd as well by letting n =2k+1