Set Theory Basics

A set is simply a group of objects or numbers. For example, {1,2,3} is a set containing the numbers 1, 2, and 3. When something belongs to a set, we say it's an "element of" that set, written with the symbol ∈. So "3 ∈ {1,2,3}" means 3 is an element of the set {1,2,3}.

When one set is contained within another, it's called a subset. For example, {1,2} is a subset of {1,2,3}. If a subset doesn't contain all elements of the original set, it's a proper subset (shown as ⊂). If it could be the entire set itself, we use ⊆. The universal set contains all elements being considered, while the empty set $∅ or {}$ has no elements.

Remember this! The empty set is NEVER written as {0} - that would be a set containing the number zero, which is different from a set with nothing in it.

Sets can be described in two main ways. The rule method describes what elements belong in the set, like {x: x > 3, x is a whole number} which means all whole numbers greater than 3. The roster method simply lists all elements, like {1,2,3}. When sets interact, we can find their union (∪) by combining all elements from both sets, or their intersection (∩) which contains only elements that appear in both sets.