Define $\sim$ on $\Bbb Z$ by $$a \sim b\longleftrightarrow a + 3b \text{ is a multiple of } 4$$
a) Use the definition of $\sim$ to show that $\sim$ is an equivalence relation on $\Bbb Z$.
b) List three integers in the equivalence class $[1]$.
For the first one, I feel like I need to prove the properties of an equivalence relation, the prompt sounds like it is for sure an equivalence relation. When I try to prove it through it's properties, the reflexive property fails, the symmetric property fails, and the transitive property is the one I am unsure of.
For the second one, I'm not sure if it has anything to do with the original prompt.
edit: My mistake on the first property. I see that it holds once I took a look it again. The symmetric property fails when I use $a = 1, b = 2$.
For the reflexive property, to check that $a\sim a$, we need that $a+3a$ is a multiple of $4$. It should be clear that this works.
For symmetry, you mention in your question the pair a=1, b=2. However, in that case, neither $a+3b=7$ nor $b+3a=5$ is a multiple of $4$, so that example doesn't break symmetry. We need to assume that $a+3b$ is a multiple of $4$, and then check that $b+3a$ is one as well. Indeed, if $a+3b$ is a multiple of $4$, then multiplying by $3$, so is $3a+9b$. However, $8b$ is also a multiple of $4$, so the difference $(3a+9b)-(8b)=b+3a$ is one as well.
For transitivity, you will want to assume that both $a+3b$ and $b+3c$ are multiples of $4$, and then show that $a+3c$ is a multiple of $4$ as well. Can you find a way to do that, perhaps getting an idea from the above proof of symmetry?
Finally, to find numbers equivalent to $1$, we just need numbers $b$ such that $1+3b$ gives us multiples of $4$. You can probably do this by trying different values for $b$, and then you might notice a pattern.
Does this help?