Set Notation and Properties

Sets help us group objects together in mathematics. We represent a set with a capital letter and list elements inside curly braces (the roster method). For example, W = {Monday, Tuesday, Wednesday, Thursday, Friday} shows a set of weekdays.

We can also use set-builder notation to define sets: W = {x | x is a weekday}, which reads "W is the set of all elements x such that x is a weekday." When a set has no elements, we call it the empty set, written as {} or ∅.

The cardinal number of a set, written as n(W), tells us how many distinct elements are in the set. Sets are equivalent if they contain the same number of elements and equal if they contain exactly the same elements (regardless of order).

Quick Tip: Remember the difference between set membership symbols: ∈ means "is an element of" while ∉ means "is not an element of." For example, Monday ∈ W but Saturday ∉ W.

Sets can be finite (with a countable number of elements) or infinite $never-ending$. A set A is a subset of set B if every element in A is also in B. It's a proper subset if A is a subset of B but A ≠ B. The empty set is a subset of every set!

For a set with n elements, there are 2^n possible subsets and 2^n - 1 proper subsets. Sets are disjoint if they share no common elements. The universal set (U) contains all elements being considered in a particular context.