Sequence

[Jarrey]

No, odd numbers are cool.  Here is something to think about:

Are the odd numbers closed* for addition (of two terms)?

Are odd numbers closed for multiplication (of two factors)?

Subtraction (two terms)?

Whether they are or are not, justify that decision in a way that should convince a fair minded listener.


*I.E., when performing those operations must you always get another odd number as a result?

Adding to odd numbers will always give an even number.    (NB any odd is always 1 more(or less) than an even number)
Let X = x+1, & Y = y+1   where x,y are even and X,Y are obviously odd, then
X+Y= x+1+y+1= x+y+2 which is the addition of 3 even numbers which is always even.
So, the odds are not closed under addition

X times Y = (x+1)*(y+1)=xy+x+y+1, which is always odd because xy,x,y are all even which adds up to an even, then +1 makes it odd
So, the odds are closed under multiplication

X - Y= (x+1) - (y+1)= x-y    which is always even since x & y are even.Even numbers are always separated by a factor of 2
So, the odds are not closed under subtraction




(NB if anyone read this earlier , I corrected the error..I hope)