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Types of Numbers

Integers	Rational	Irrational	Complex
$\{\dots,-3,-2,-1,0,1,2,3,\dots\}$	$\frac{a}{b},\; b\ne0,\; a,b\in\mathbb{Z}$	A number that cannot be expressed as a fraction, e.g. $\pi$	$x=a+bi$ where $a$ and $b$ are any real numbers

Reflexive	Symmetric	Transitive	Substitution
$a=a$	$a=b \Rightarrow b=a$	$a=b \text{ and } b=c \Rightarrow a=c$	$a=b \Rightarrow b \text{ can replace } a$

Both $A$ and $B$ have the same elements $A=B$

Subset: If every element of $A$ is in $B$ $A\subseteq B \Rightarrow A\subseteq B \land B\subseteq A \Leftrightarrow A=B$

Proper Subset: If every element in $A$ is also in $B$ but $A\ne B$ $A\subset B$

Intersection: The elements that are both in $A$ and in $B$ $A\cap B=\{x|x\in A \land x\in B\}$

Union: All elements from $A$ and $B$ $A\cup B=\{x|x\in A \lor x\in B\}$

Complement: If $A\subset U$ , and $U$ is the universal set $\overline{A}=A^c=\{x|x\in U \land x\notin A\}$

a. $U^c=\varnothing$ b. $\varnothing^c=U$ c. $(A^c)^c=A$ d. $A\cup A^c=U$ e. $A\cap A^c=\varnothing$

$A\cup B=B\cup A$	Commutative law for union
$A\cap B=B\cap A$	Commutative law for intersection
$A\cup(B\cup C)=(A\cup B)\cup C$	Associative law for union
$A\cap(B\cap C)=(A\cap B)\cap C$	Associative law for intersection
$A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$	Distributive law for union
$A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$	Distributive law for intersection