Q:

Is this proof on addition valid

this proof has to show that adding an even number and an odd number give an odd number. So let $a$ be an even number and $b$ is an odd number $c$ is an even number and $c=b-1$ so:

$$a+b=d$$ substitute $b=c+1$ so, $$a+c+1=d$$ use the fact that $even+even=even$ so $a+c$ is even and $even+1=odd$ so $d$ is odd. Hence $$even+odd=odd$$ Is this proof valid?

A:

Yes, well, mostly.   You have shown that $d$ is the addition of $1$ to the sum of two even numbers (from the premises), and claim by some facts that infers that $d$ is an odd number.  
Your argument is valid if you are given those facts.   Were they provided as part of the problem?   Otherwise you will have to demonstrate them from the definitions of even and odd numbers.

