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Let $A$ be a nonempty set and let $B$ be a fixed subset of $A$. A relations $R$ is defined on the power set $\mathcal{P}(A)$ by $X\mathrel{R}Y$ if $X \cap B = Y \cap B$.

Let $A=\{1,2,3,4,5\}$ and $B=\{1,3\}$. For a subset $X=\{2,3,4\}$, Determine the equivalence class $[X]$?

I have never heard of something like this. Any ideas to get started?

Brian M. Scott
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Mathgirl
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1 Answers1

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Note $[X]=\{Y \subset A \mid X\mathrel{R}Y \}$. You have that $X \cap B=\{3\}$, thus \begin{equation*} [X]=\{Y \subset A \mid Y \cap \{1,3\} = \{3\}\} \end{equation*} What other subsets of $A$ can you construct that intersect with $B$ to get $\{3\}$? Some examples include $\{3,4\}$ and $\{2,3,5\}$.

Brian M. Scott
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Sloan
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  • Do you mean like a subset of A could be {1,3} and intersect with B to get {3}? How would this relate to being an equivalence class [X]? – Mathgirl Apr 29 '15 at 16:41
  • For example, $Y={3,4}$ has the property that $X \cap B = {3} = Y \cap B$, so $Y \in [X]$. – Sloan Apr 29 '15 at 16:42
  • So any element of A paired with 3 will be apart of [X]? – Mathgirl Apr 29 '15 at 16:45
  • No, ${1,3} \cap B \neq {3}$, so this set is not a member of $[X]$. I've edited my answer to hopefully provide more clarity. – Sloan Apr 29 '15 at 16:46