## lgbasallote 3 years ago Prove that if n is an integer then 3n + 2 is even, then n is even

1. mathslover

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

2. vf321

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

3. mathslover

ok , thanks

4. vf321

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

5. mathslover

if 3n is even then n is even.

6. sara12345

Given : 3n+2 is even To prove : n is even 3n+2 = 2k 3n = 2k-2 3n = 2(k-1)

7. hartnn

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

8. vf321

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

9. mathslover

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

10. vf321

But it's not so no guarentee is made.

11. hartnn

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

12. mathslover

@lgbasallote ?

13. JamesWolf

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

14. sara12345

yes

15. sara12345

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

16. JamesWolf

right I see

17. vf321

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

18. lgbasallote

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

19. hartnn

but whats the question ?

20. lgbasallote

the statement in the blue box

21. lgbasallote

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

22. lgbasallote

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

23. JamesWolf

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

24. hartnn

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

25. lgbasallote

if you use direct proof... it should be 3n + 2 = 2x then prove n is even

26. lgbasallote

but like i said...should be contradiction though

27. vf321

well okay thats easy enough, too

28. lgbasallote

hmmm then let's see you try

29. vf321

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

30. sara12345

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

31. vf321

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

32. lgbasallote

33. sara12345

we do that always in proof by contradiction

34. sara12345

RHS = even , LHS = odd => contradiction

35. lgbasallote

i was referring to your solution

36. vf321

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

37. JamesWolf

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

38. lgbasallote

hmm seems legit

39. lgbasallote

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

40. vf321

As @JamesWolf showed for you,

41. vf321

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

42. JamesWolf

no its not legit

43. JamesWolf

ive been an idiot int he first step

44. vf321

@JamesWolf no, you inadvertently put up my proof.

45. sara12345

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

46. lgbasallote

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

47. vf321

@sara12345 yes you're right, sorry.

48. JamesWolf

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

49. lgbasallote

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

50. lgbasallote

anyway...is my proof right?

51. sara12345

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