# Defining an equivalence relation by X~Y iff X,Y have the same number of elements

Let A=P({1,2,3,4}). Notice that A has 16 elements (all subsets of a 4-element set). Defining an equivalence relation by X~Y iff X,Y have the same number of elements, here are two equivalence classes: [∅] = { ∅ } [{1}] = { {1}, {2}, {3}, {4} } Find and list the remaining ones.

this is what I got

For the congruence modulo 2, there are 2 equivalence classes: [0] = all even numbers, and [1] = all odd numbers. 2. For congruence mod 3, there are 3 equivalence classes: [0], [1], and [2].

• What is challenging you about this? Can you list all two-element subsets of $\{1,2,3,4\}$? Can you list all three-element subsets? How about four-element subsets? That is all they are asking you to do here... Commented Apr 18, 2020 at 16:59
• The is not about "congruence modulo $2$" or "congruence modulo $3$", this is instead about counting: count the number of subsets with $2$ elements; count the number of subsets with $3$ elements. Commented Apr 18, 2020 at 17:37

$$[X] = \{Y\in \mathcal{P}(A) : |Y| = |X|\}$$.
$$[\emptyset] = \lbrace \emptyset \rbrace$$ $$[\lbrace 1 \rbrace] = \lbrace \lbrace 1 \rbrace, \lbrace2\rbrace,\lbrace 3\rbrace, \lbrace4\rbrace \rbrace$$ $$[\lbrace 1,2 \rbrace]=\{ \lbrace 1,2 \rbrace,\lbrace 1,3 \rbrace,\lbrace 1,4\rbrace,\lbrace 2,3 \rbrace, \lbrace 2,4 \rbrace, \lbrace 3,4 \rbrace \}$$ $$[\lbrace 1,2,3 \rbrace]=\{ \lbrace 1,2,3 \rbrace,\lbrace 1,2,4 \rbrace,\lbrace 1,3,4\rbrace,\lbrace 2,3,4\}$$ $$[A] = \lbrace A \rbrace$$