While proving the injectivity of a function, while proving x1=x2 from f(x1) = f(x2); if f(x) has two different values for two different conditions (let's say for x=odd and x=even), do we have to take all the combinations?
x1 and x2 are both even
x1 and x2 are both odd
x1 is even and x2 is odd
x1 is odd and x2 is odd

And in the last two cases, do we have to prove x1 not equal to x2?

Question

While proving the injectivity of a function, while proving x1=x2 from f(x1) = f(x2); if f(x) has two different values for two different conditions (let's say for x=odd and x=even), do we have to take all the combinations?
x1 and x2 are both even
x1 and x2 are both odd
x1 is even and x2 is odd
x1 is odd and x2 is odd

And in the last two cases, do we have to prove x1 not equal to x2?

myininaya · Answer

Is this the question that was given to you?

dustin_whitelock · Answer

No, this is my doubt.

myininaya · Answer

Can you tell me what problem you were looking at that gave you this doubt?

dustin_whitelock · Answer

Actually if I take all the cases, then the last two cases are proved not true. By logic of injectivity all the cases have to be true, so I was wondering if the last two cases have to be proved not to be true.

dustin_whitelock · Answer

I'll post the question right away.

dustin_whitelock · Answer

attachment

dustin_whitelock · Answer

Now, this website says...



I'm not sure why we should rule out the possibility.

dustin_whitelock · Answer

Anyone replying?

myininaya · Answer

ok let's start there 
say we have n is odd and m is even just like that link says
and then we have n-1=m+1
implies n-m=2

can an even and odd number ever by 2 units away from each on the number line

myininaya · Answer

|dw:1466271009693:dw|
notice even numbers can be 2 units apart (or even number of units apart)
notice odd numbers can be 2 units apart (or even number of units apart)
but
a pair of odd and even number cannot be 2 units apart