Set theory is the branch of mathematics that studies sets, which are collections of distinct objects.
Member (∈): If o is a member (or element) of A, we write o∈A. The key relation between sets is membership – when one set is an element of another. If a is a member of B, this is denoted a ∈ B, while if c is not a member of B then c ∉ B. For example, with respect to the sets A = {1,2,3,4}, B = {blue, white, red}, and F = {n2 − 4 : n is an integer; and 0 ≤ n ≤ 19} defined above,
4 ∈ A and 285 ∈ F; but
9 ∉ F and green ∉ B.
4 ∈ A and 285 ∈ F; but
9 ∉ F and green ∉ B.
Subset relation or set inclusion (⊆): A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1,2} is a subset of {1,2,3} , but {1,4} is not.
If every member of set A is also a member of set B, then A is said to be a subset of B, written A ⊆ B (also pronounced A is contained in B). Equivalently, we can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by ⊆ is called inclusion or containment.
If A is a subset of, but not equal to, B, then A is called a proper subset of B, written A ⊊ B (A is a proper subset of B) or B ⊋ A (B is a proper superset of A).
Note that the expressions A ⊂ B and B ⊃ A are used differently by different authors; some authors use them to mean the same as A ⊆ B (respectively B ⊇ A), whereas other use them to mean the same as A ⊊ B (respectively B ⊋ A). A is a subset of B
Example:The set of all men is a proper subset of the set of all people.
{1, 3} ⊊ {1, 2, 3, 4}.
{1, 2, 3, 4} ⊆ {1, 2, 3, 4}.
{1, 3} ⊊ {1, 2, 3, 4}.
{1, 2, 3, 4} ⊆ {1, 2, 3, 4}.
The empty set is a subset of every set and every set is a subset of itself:∅ ⊆ A.
A ⊆ A.
A ⊆ A.
An obvious but useful identity, which can often be used to show that two seemingly different sets are equal:
A = B if and only if A ⊆ B and B ⊆ A.
A = B if and only if A ⊆ B and B ⊆ A.
Union (∪): Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4}.
Intersection (∩): Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B. The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3} .
Complement: Complement of set A relative to set U, denoted Ac, is the set of all members of U that are not members of A. This terminology is most commonly employed when U is a universal set, as in the study of Venn diagrams. This operation is also called the set difference of U and A, denoted U \ A. The complement of A: {1,2,3} relative to U: {2,3,4} is {4} , while, conversely, the complement of {2,3,4} relative to {1,2,3} is {1} .
Symmetric difference : Symmetric difference of sets A and B is the set of all objects that are a member of exactly one of A and B (elements which are in one of the sets, but not in both). For instance, for the sets {1,2,3} and {2,3,4} , the symmetric difference set is {1,4} . It is the set difference of the union and the intersection, (A ∪ B) \ (A ∩ B).
Cartesian(x) : Cartesian product of A and B, denoted A × B, is the set whose members are all possible ordered pairs (a,b) where a is a member of A and b is a member of B.
Power set: Power set of a set A is the set whose members are all possible subsets of A. For example, the powerset of {1, 2} is { {}, {1}, {2}, {1,2} } .
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