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Intersection of Sets

The intersection of sets A and B is the set of elements which are common to A and B, that is, those elements which belong to both A and B.

We denote the intersection of A and B by A  B which is read as "A intersection B".

In the following figures A
 B is given by in the Venn diagram. The shaded portion represents A  B. Note that A  B consists of the area common to both A and B.

If A and B have no common elements, A  B = φ, then A and B are said to be disjoint.
Illustration 1
Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. How many elements are there in A intersection B, and what are they?

Solution
There are two elements in A  B. The elements are 3 and 4.

Illustration 2
Let A = {a, b, c, d}, B = {c, d, e, f}, and C = {d, e, f, g}, then
A  B = {c, d}, B  C = {d, e, f}, A  C = {d}.
note that A  A = {a, b, c, d}= A.
In fact, A  A = A for every set A.
Also, note that A  φ = φ for every set A.

Illustration 3
Let A = {1, 2, 3, 4} and B = {5, 6, 7}, then A  B = φ because A and B have no elements in common. Such sets are said to be disjoint sets.

Two sets A and B are said to be disjoint, if A  B = φ. Two sets are called disjoint if they have no elements in common, that is, if their intersection is disjoint. Two disjoint events are also called mutually exclusive events.
If E denotes the set of even numbers and O denotes the set of odd numbers, their intersection is disjoint. It is shown by the diagram given below:

The set E and O are disjoint, as they have no elements in common.

Example
Find the intersection of the following sets
  1. A = {a, e, i, o, u}; B = {a, b}
  2. A = {x : x is a natural number which is a multiple of 3};
  3. B = {x : x is a natural number and x 25}
  4. A = {x : x is a natural number and 1 < x < 6};
  5. B = {x : x is a natural number and 6 < x < 10}
  6. A = {1, 2, 5, 7, 11}; B = φ
  7. A = {1, 2, 5, 7, 11}; B = {5, 7, 11, 13, 19}
  8. A = {a, b, c}; B = {d, e, f, g}
Solution
  1. A B = {a}    
  2. A B = {3, 6, 9, 12, 15, 18, 21, 24}
  3. A B = φ     
  4. A B = φ   
  5. A B = {5, 7, 11}
  6. A B = φ.




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