Universal set and Venn diagrams
The universal set, written ξ, contains every element being considered in a problem. A Venn diagram shows sets as circles drawn inside a rectangle; the rectangle stands for the universal set ξ and each circle is a set within it.
Key idea / 要点
The complement of A, written A′, is the set of all elements in ξ that are not in A. So n(A′) = n(ξ) − n(A). A is a subset of B (A ⊆ B) if every element of A is also in B.
Worked example
Worked example / 例题
Let ξ = {1, 2, 3, 4, 5, 6, 7, 8} and A = {1, 2, 3}.
The complement A′ = {4, 5, 6, 7, 8}, so n(A′) = 5.
Check: n(ξ) − n(A) = 8 − 3 = 5. ✓
Subsets
Every element of a subset must belong to the bigger set, so {1, 2} ⊆ {1, 2, 3, 4}. The empty set ∅ is a subset of every set, and every set is a subset of itself. A set with n elements has 2ⁿ subsets in total. For example, {a, b} has 2² = 4 subsets: ∅, {a}, {b}, {a, b}. A subset that is not the whole set itself, such as {a}, is called a proper subset. When you read a Venn diagram, a circle drawn completely inside another circle shows that one set is a subset of the other.
Remember / 记住
- ξ is the universal set; the rectangle in a Venn diagram.
- A′ is everything in ξ that is not in A; n(A′) = n(ξ) − n(A).
- A set with n elements has 2ⁿ subsets.