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Union of Set

Let A and B be two sets. The union of A and B is the set of all the elements which belong to A or to B or to both.

The word union means "united" or "all together." When you combine the elements of set A and the elements of set B, such that the common elements are taken only once to form a new set called the union of A and B and denoted by
A  B

Here in this problem as the car appears in the both the sets, it has to appear only once in A  B.
Thus, A  B contains 5 items only.

In the Venn diagram, A  B is given by figures given below. The shaded portion represents A  B. Note that it consists of the area of A and area of B.
 

The union of two sets A and B may also be defined concisely by
A B = {x : x A or x B}
Note that both A and B are subsets of A  B that is, A  A  B and B  A  B.

 

Illustration 1
Let A = {1, 2, 3, 4, 5, 6} and B = {5, 6, 8, 9, 11} then
A
 B = {1, 2, 3, 4, 5, 6, 8, 9, 11}.

Illustration 2
Let A = {1, 2, 3, 4} B = {2, 4, 6, 8,} and C = {3, 4, 5, 6}.
We shall find A
 B, B  C, A  C and B  B.
A
 B = {1, 2, 3, 4, 6, 8},  B  C = {2, 3, 4, 5, 6, 8}
A
 C = {1, 2, 3, 4, 5, 6} and B  B = {2, 4, 6, 8}.
Note that B
 B = B.
In fact, for every set A, A
 A = A

Illustration 3
Let A = {a, b, c} and B = {b, c, d, e}.
We have A
 B = {a, b, c, d, e} and B  A = {b, c, d, e, a}
Note that A
 B = B  A, since A  B and B  A consists of the same elements.
In general, for any two sets A and B.
A
 B = B  A.

This shaded region indicates A  B.
Let us now shade B
 A.

This violet shaded portion indicates B  A. In both the diagrams the shaded region remains the same. Thus we conclude that A  B = B  A. 

Illustration 4
Let A = {a, b, c}. What is the union of A  φ ? 
By definition A  φ consists of all the elements which belong to A or to φ . Since φ contains no element, A  φ consists of the elements of A only. Thus A  φ = A. This is true for all the sets A.

Example
For the following sets, find their union
  1. A = {a, b, c, d}; B = {c, d, e, f}
  2. A = {x : x N}; B = {x : x x}
  3. A = {x  x is an even natural number};
    B = {x  x is an odd natural number}.
  4. A = {x : x is a rational number};
    B = {x : x is an irrational number}.
  5. A = {x : x is a vowel of the English alphabet};
    B = {x : x is a consonant of the English alphabet}.
Solution
  1. {a, b, c, d, e, f}       
  2. {x : x N}
  3. {x : x N}          
  4. {x : x R}
  5. {x : x is a letter of the English alphabet}




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