Question

Let X = {1,2,3}. Define a relation ∼ on P(X) by A ∼ B if A and B have the same number of elements.

Prove that ∼ is an equivalence relation and write down all equivalence classes of ∼.

Answer 1

Answer on Question #76255 – Math – Discrete Mathematics.

Question

Let $X = \{1,2,3\}$. Define a relation $\sim$ on $P(X)$ by $A \sim B$ if $A$ and $B$ have the same number of elements.

Prove that $\sim$ is an equivalence relation. Write down all equivalence classes of $\sim$.

Solution

**Reflexively:** Let $A \in P(x)$. Then $A \sim A$ because the set has an unaltered number of elements.

**Symmetric:** Let $A \in P(x)$, $B \in P(x)$ and $A \sim B$. Since $A \sim B$, $|A| = |B|$. Then $|B| = |A|$. Since $|B| = |A|$, $B \sim A$.

**Transitivity:** Let $A \in P(x)$, $B \in P(x)$, $C \in P(x)$, $A \sim B$ and $B \sim C$. Since $A \sim B$, $|A| = |B|$. Since $B \sim C$, $|B| = |C|$. Then $|A| = |C|$ because $|A| = |B|$ and $|B| = |C|$. Since $|A| = |C|$, $A \sim C$.

Equivalence classes of $\sim$:

$$[\varnothing] = \{\varnothing\} \text{ (zero of elements)},$$ $$[\{1\}] = \{\{1\}, \{2\}, \{3\}\} \text{ (one element)},$$ $$[\{1,2\}] = \{\{1,2\}, \{1,3\}, \{2,3\}\} \text{ (two elements)},$$ $$[\{1,2,3\}] = \{\{1,2,3\}\} \text{ (three elements)}.$$

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