Basic Mathematical Notation

Sets §

• collection of objects with some common property
• if $x$ is an element in $X$, we write $x\in X$.
• if $x$ is not in $X$, we write $x\notin X$.
• if $x_1,x_2,x_3$ are the elements of $X$, $X=\{x_1,x_2,x_3\}$.
• $\{\}=\varnothing =\varnothing$ is empty set, or set with no elements.

Examples §

• $\mathbb{N}=\{1,2,3,\dots \}$ is natural numbers
• $\mathbb{Z}=\{0,1,-1,-2,\dots \}$ is integers
• $\mathbb{Q}=\left\{\frac{a}{b}|a,b\in \mathbb{Z},b\ne 0\right\}$ is rational numbers
• $\mathbb{R}$ is real numbers.

Combining/Comparing Sets §

• union: $A\cup B=\{x|x\in A\text{ or }x\in B\}$.
• intersection: $A\cap B=\{x|x\in A\text{ and }x\in B\}$ .
• $A\subset B$ if every element of $A$ is also in $B$.

Examples §

• $A\cap B\subset A$
• $B\subset A\cup B$.
• $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}\subset \mathbb{C}$.