Question

f ~ g <-> {x in N | f(x) not equal g(x)} is finite.
show that this is an equivalence relation
Help with transitive proof pls!!!!

Answer 1

so you have to show if \(\{{x\in \mathbb{N}|f(x)\neq g(x)}\}\) is finite, and \(\{x\in \mathbb{N}|g(x)\neq h(x)\}\) is finite, then \(\{x\in \mathbb{N}|f(x)\neq h(x)\}\) is finite
i am a bit tired, but isn't it the largest that set can be is the union of the first two?

Answer 2

but im having problems proving that if f(x) not equal to g(x), and g(x) not equal to h(x) that f(x) no equal to h(x), just because they don't equal g(x) doesn't mean they are not equal one another...

Answer 3

we need to assume that f(x) is not equal to h(x) and then prove the set is finite... but i don't know how to do that :/