Set

Definition

• An unordered collection of distinct objects (elements)
• A set is said to contain its elements.
  • Set with no element is empty set or null set
  • A set with one element is called a singleton set
• We write $a\in A$ to denote that $a$ is an element of the set $A$
  • For every set, $\varnothing \subseteq S$ and $S\subseteq S$ theorem

Ways to specify a set:

• set roster: $\{a,b,c\}$
• set builder notation: $P=\{x\in {\mathbb{Z}}^{+}|x\text{ is prime nb and }x<20\}$: $x\in {\mathbb{Z}}^{+}$ such that …
• Popular sets

Popular sets:

• $(\mathbb{Q}=\{p/q|p\in \mathbb{Z},q\in \mathbb{Z},\text{and }q\ne 0\})$
• | Natural | Integer | Positive int | Rational | Real | Positive Real | Complex | | --- | --- | --- | --- | --- | --- | --- | | $\mathbb{N}$ | $\mathbb{Z}$ | ${\mathbb{Z}}^{+}$| $\mathbb{Q}$ | $\mathbb{R}$ | ${\mathbb{R}}^{+}$| $\mathbb{C}$|

Equal sets: $A=B$

• Two sets are equal if and only if they have the same elements.
  • We write $A=B$ if $A$ and $B$ are equal sets.

Subset: $A\subset B$

• A is subset of B, B is super set of A $(B\supseteq A)$
  • $A\subset B$: proper subset, when $A\ne B$
  • if $A\subseteq B$ and $B\subseteq A⟹A=B$

Set cardinality:$|S|$

• $|S|$: non-negative integer denotes exact number of element for a finite set S
  • 1. $|S|=n\to |P(S)|=2^n$

Infinite set:

• A set is said to be infinite if it is not finite

Power sets: $\mathcal{P}(S)$

• power set of S is set of all subsets of set S, e.g. $\mathcal{P}(1,2)=\{\varnothing ,\{1\},\{2\},\{1,2\}\}$
  • $\mathcal{P}(\varnothing )=\{\varnothing \}$
  • $\mathcal{P}(\{\varnothing \})=\{\varnothing ,\{\varnothing \}\}$
  • every subset of a set is an element in its power set

Set operations:

• Union operation
• Intersect operation
• Difference operation
• Complement operation