A is a subset of B when every element of A is in B. A set with n elements has 2ⁿ subsets, collected in its power set. Intervals are subsets of the real number line written with brackets.
Every vowel is a letter of the alphabet. So every element of the set of vowels is also an element of the set of letters.
A is a subset of B, written A ⊆ B, if every element of A is also an element of B. {a, e} ⊆ {a, e, i, o, u}. If at least one element of A is not in B, then A is not a subset of B, written A ⊈ B.
Every set is a subset of itself: A ⊆ A, because every element of A is in A.
The empty set is a subset of every set: ∅ ⊆ A. To show A ⊈ B, you must find an element of A that is not in B. The empty set has no elements, so no such element can be found. Therefore ∅ ⊆ A for every A.
Why is ∅ a subset of a set it shares no elements with, such as {1, 2}?
A ⊆ B fails only if some element of A is not in B.
∅ has no elements, so there is no element of ∅ that could be missing from {1, 2}. The condition “every element of ∅ is in {1, 2}” has no cases to check, and so it is true.
This follows directly from the definition. It is also needed for the count of subsets below to be correct.
A is a proper subset of B, written A ⊂ B, if A ⊆ B and A ≠ B. Then B has at least one element that A does not.
{a, e} ⊂ {a, e, i, o, u}. But {a, e, i, o, u} is not a proper subset of itself, because it is equal to itself.
The power set of A, written P(A), is the set of all subsets of A. For A = {1, 2}, the subsets are ∅, {1}, {2} and {1, 2}. So P(A) = { ∅, {1}, {2}, {1, 2} }, which has 4 elements.
A set with n elements has exactly 2ⁿ subsets. To build a subset, decide for each element whether it is in or out. That is two choices per element, and n independent choices give 2 × 2 × … × 2 = 2ⁿ subsets. The choice “all out” gives ∅, and “all in” gives A itself.
How many subsets does {a, b, c} have? List them.
2³ = 8.
∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}.
Each of the three elements is in or out: 2 × 2 × 2 = 8 combinations, one subset for each.
An interval is the set of all real numbers between two endpoints. It is a subset of R, the set of real numbers. Brackets show whether each endpoint is included.
[2, 5] contains 2, 5 and every real number between them. (2, 5) contains every real number between 2 and 5, but not 2 or 5. Both are infinite sets.
Endpoints in or out. A square bracket and a filled dot mean the endpoint is included. A round bracket and a hollow dot mean it is excluded.
1A ⊆ B means…
2Which statement is true for every set A?
3How does A ⊂ B differ from A ⊆ B?
4A set with n elements has ____ subsets.
5How many subsets does {a, b, c} have?
6How does [2, 5] differ from (2, 5)?
Next: the universal set, and Venn diagrams.
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