## Answers

Combining all the elements of any two sets is called the Union of those sets

Union of two sets A and B is obtained by combining all the members of the sets and is represented as A ∪ B

More About Union of SetsIn the union of sets, element is written only once even if they exist in both the sets.Union of two sets is commutative i.e. if A and B are two sets, then A ∪ B = B ∪ AUnion of sets is also associative. If A, B, and C are three sets, then A ∪ (B ∪ C) = (A ∪ B) ∪ CExamples of Union of SetsIf A = {1, 2, 3, 4, 5} and B = {2, 4, 6}, then the union of these sets is A ∪ B = {1, 2, 3, 4, 5, 6}

Basic operations[edit]

There are several fundamental operations for constructing new sets from given sets.

Unions[edit]The union of A and B, denotedA ∪ BMain article: Union (set theory)Two sets can be "added" together. The union of A and B, denoted by A ∪ B, is the set of all things that are members of either A or B.

Examples:

{1, 2} ∪ {1, 2} = {1, 2}.{1, 2} ∪ {2, 3} = {1, 2, 3}.{1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}Some basic properties of unions:

A ∪ B = B ∪ A.A ∪ (B ∪ C) = (A ∪ B) ∪ C.A ⊆ (A ∪ B).A ∪ A = A.A ∪ ∅ = A.A ⊆ B if and only if A ∪ B = B.Intersections[edit]Main article: Intersection (set theory)A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things that are members of both A and B. If A ∩ B = ∅, then A and B are said to be disjoint.

The intersection of A and B, denoted A ∩ B.Examples:

{1, 2} ∩ {1, 2} = {1, 2}.{1, 2} ∩ {2, 3} = {2}.Some basic properties of intersections:

A ∩ B = B ∩ A.A ∩ (B ∩ C) = (A ∩ B) ∩ C.A ∩ B ⊆ A.A ∩ A = A.A ∩ ∅ = ∅.A ⊆ B if and only if A ∩ B = A.Complements[edit]The relative complementof B in AThe complement of A in UThe symmetric difference of A andBMain article: Complement (set theory)

Two sets can also be "subtracted". The relative complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B (or A − B), is the set of all elements that are members of A but not members of B. Note that it is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect.

In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′.

Examples:

{1, 2} \ {1, 2} = ∅.{1, 2, 3, 4} \ {1, 3} = {2, 4}.If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then U \ E = E′ = O.Some basic properties of complements:

A \ B ≠ B \ A for A ≠ B.A ∪ A′ = U.A ∩ A′ = ∅.(A′)′ = A.A \ A = ∅.U′ = ∅ and ∅′ = U.A \ B = A ∩ B′.An extension of the complement is the symmetric difference, defined for sets A, B as

{\displaystyle A\,\Delta \,B=(A\setminus B)\cup (B\setminus A).}For example, the symmetric difference of {7,8,9,10} and {9,10,11,12} is the set {7,8,11,12}.

Cartesian product[edit]Main article: Cartesian productA new set can be constructed by associating every element of one set with every element of another set. The Cartesian product of two sets A and B, denoted by A × B is the set of all ordered pairs (a, b) such that a is a member of A and b is a member of B.

Examples:

{1, 2} × {red, white} = {(1, red), (1, white), (2, red), (2, white)}.{1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green) }.{1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.Some basic properties of cartesian products:

A × ∅ = ∅.A × (B ∪ C) = (A × B) ∪ (A × C).(A ∪ B) × C = (A × C) ∪ (B × C).Let A and B be finite sets. Then

| A × B | = | B × A | = | A | × | B |.For example,

{a,b,c}×{d,e,f}={(a,d),(a,e),(a,f),(b,d),(b,e),(b,f),(c,d),(c,e),(c,f)}.