Sets

• Sequence cannot matter
• No multiplicity (things can only appear once)

Carinality: The number of elements in the set

Notation

• “Roster Notation” : use a pattern that expands to the entire set {1, 2, 3, …}
• $|A|$ : Cardinality AKA number of elements
• “Set Builder Notation” : Prototype : Condition {$n$ : $n\in \mathbb{N}\&n\le 5$}
• A is a subset of B: $A\subset B$, Or including equality (They are the same set): $A\subseteq B$

Intervals [a,b] = {x:x \in R and a⇐x⇐b} (a,b) = {x:x \in R and a⇐x⇐b}

Special Sets

• $\mathbb{N}$ : All natural numbers {1, 2, 3, …}
• $\mathbb{Z}$ : All integers {…, -2, -1, 0, 1, 2, …}
• $\mathbb{Q}$ : Rational Numbers {$\frac{a}{b}$ : $a\in \mathbb{Z},b\in \mathbb{N}$} (Incomplete)
• $\mathbb{R}$ : Real numbers
• $0$ : Empty set {}
• Sets that contain other sets: {1, 2, {1, 2}}

Power Set The set consisting of all possible combinations of all elements of a set. $\mathcal{P}(A)=\{S:S\le A\}$ “The set of all subsets of A” $|\mathcal{P}(A)|=2^{|A|}$

Operations

Cartesian Product $A\times B=\{(a,b):a\in A,b\in B\}$ Set of all the possible ordered pairs. $|A\times B|=|A|\times |B|$ ${\mathbb{R}}^2=\mathbb{R}\times \mathbb{R}$

Compliments $\mathbb{U}$ : Universal set. Contains “Everything”, but taken to mean whatever is relevant based on context. $\bar{A}=\{x:x\in U-A\}$

De Morgans’ Laws $\overline{A\cap B}=\overline{A}\cup \overline{B}$ $\overline{A\cup B}=\overline{A}\cap \overline{B}$ $\neg (x\in N.g(x))\to \exists x\in N.\neg g(x)$

$\forall x\in \mathbb{N}g(x)$: For all x in N, g(x) $\exists x\in S.P(x)$: There exists an x an S such that p(x)