6. Equality of sets, powersets

Equality of sets

How to prove $A=B$ or $A\ne B$

$\boxed{A=B}$ means that the sets $A$ and $B$ have the same elements. This is the statement of the form $\forall x(x\in A⟺x\in B)$.

$$A=B⟺A\subseteq B\text{ and }B\subseteq A$$ $$A\ne B⟺A⊈B\text{ and }B⊈A$$

If we need to show that $\boxed{A=B}$ then:

• We need to show both $\boxed{A\subseteq B}$ and $\boxed{B\subseteq A}$.

If the task is to show that $\boxed{A\ne B}$ then:

• We either need to find an element in $A$ that is not in $B$ or an element in $B$ that is not in $A$.

Powersets

Power set

The power set, $\boxed{P(S)}$, is the set of all subsets of a set $S$. So $P(S) = \{ A | A \subseteq S \}$.

As every set $S$ will always have $\varnothing \subseteq S$ and $S\subseteq S$, then $\boxed{\varnothing \subseteq P(S)}$ and $\boxed{S\subseteq P(S)}$ will always be true.

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Examples:

• $P(\{u,v,w\})=\{\varnothing ,\{u\},\{v\},\{w\},\{u,v\},\{v,w\},\{w,u\},\{u,v,w\}\}$
• $P(\{\text{Among, Us}\})=\{\varnothing ,\{\text{Among}\},\{\text{Us}\},\{\text{Among, Us}\}\}$
• $P(\{s,u,s\})=\{\varnothing ,\{s\},\{u\},\{s,u,s,u,s\}\}$
• $P(\mathbb{N})=\{X|X\subseteq \mathbb{N}\}$ We can say $\{123,96,3\}\in P(\mathbb{N})$, $\{2,81\}\in P(\mathbb{N})$ or $\varnothing \in P(\mathbb{N})$. But $\{123,-300\}\notin P(\mathbb{N})$, $\{\frac{5}{3}\}\notin P(\mathbb{N})$ and $\{\varnothing \}\notin P(\mathbb{N})$.