Relation between subsets of a power set Relations are defined on a power set $P(\Bbb N)$

*

*A~B, if A\B is finite

*A~B, if $A\Delta B$ is finite

*A~B, if $A\cap B$ is finite

Show if 1),2) and 3) are reflexive, symmetric or transitive.
First I assume that A and B are subsets of $P(\Bbb N)$ and R would be the relation between both sets...
For 1) A\B = $\{x | x \in A\text{ and }x \notin B\}$  Since xRx is not fullfilled, this is not a reflexive relation? I don't know if i understood it correctly ..If someone could quickly point it out!
Thanks
 A: This list of set identities and relations on Wikipedia could be helpful.

*

*$A \sim B$ if and only if $A \setminus B$ is finite.


*

*Reflexive ✔️ $A \setminus A = \varnothing$ is finite

*Symmetric ✘ Let $A = 4 \mathbb{N} = \{ 4, 8, 13, \ldots \}$ and $B = 2 \mathbb{N} = \{ 2, 4, 6, 8, \ldots \}.$ Then $A \setminus B = \varnothing$ but $B \setminus A = \{ 2, 6, 10, \ldots, \} = 2 + 4 \mathbb{N}.$

*Transitive ✔️ Assume $A \setminus B = \{ a_1, \ldots, a_m \}$ and $B \setminus C = \{ b_1, \ldots, b_n \}$. Want to show that $A \setminus C$ is finite. Let $x \in A \setminus C.$ If $x \not\in B$ then $x \in A \setminus B = \{ a_1, \ldots, a_m \}$ while if $x \in B$ then $x \in B \setminus C = \{ b_1, \ldots, b_n \}.$ Thus $A \setminus C \subseteq \{ a_1, \ldots, a_m, b_1, \ldots, b_n \}.$


*$A \sim B$ if and only if $A \triangle B$ is finite.


*

*Reflexive ✔️ $A \triangle A = \varnothing$ is finite

*Symmetric ✔️ Because $A \triangle B = B \triangle A$ so one side is finite if and only if the other side is finite.

*Transitive ✔️ Because $A \triangle C = (A \triangle B) \triangle (B \triangle C) \subseteq (A \triangle B) \cup (B \triangle C)$ (see list of identities here) where the right hand side is a finite set when $A \sim B$ and $B \sim C$ are true.



*$A \sim B$ if and only if $A \cap B$ is finite.


*

*Reflexive ✘ $A \cap A = A$ so if $A = \mathbb{N}$ (for example) then $A \cap A$ is not finite so $A \sim A$ is false.

*Symmetric ✔️ Because $A \cap B = B \cap A$ so one side is finite if and only if the other side is finite.

*Transitive ✘ Let $A$ be an infinite set (for example, $A = \mathbb{N}$), let $C = A,$ and $B = \varnothing.$ Then $A \cap B = \varnothing$ is finite and $B \cap C = \varnothing$ is finite but $A \cap C = A = C$ is infinite.

