OPERATION ON SET
As we all know that whenever we have gone to carry out the operation of addition, multiplication, division, and subtraction on two numbers, we get new numbers. Similarly, when we suppose to carry out the operation of Union & Intersection on two sets, we get new sets, just as we can find the difference between two numbers, we can also find the difference between two sets.
Union of Two Sets –
It has been stated that, Union of the two sets A & B is the set of all the elements that belong to either A or B or both and it is denoted by A U B ( we should read as A union B ).
We can write A U B = { x : x ∈ A or x ∈ B }
For better understanding, we would like to provide some example –
Example.1) If, A = { 2, 4, 6, 8 } and B = { 6, 8, 10, 12 } , then find A U B = ?
Ans.) As per the given condition A = { 2, 4, 6, 8 } and B = { 6, 8, 10, 12 }
So, A U B = { 2, 4, 6, 8 } U { 6, 8, 10, 12 }
= { 2, 4, 6, 8, 10, 12 }
Example.2) If, A = { 1, 2, 3, 4, 5, 6, 7, 8 } and B = { 6, 8, 10, 12 } , then find A U B = ?
Ans.) As per the given condition A = { 1, 2, 3, 4, 5, 6, 7, 8 } and B = { 6, 8, 10, 12 }
So, A U B = { 1, 2, 3, 4, 5, 6, 7, 8 } U { 6, 8, 10, 12 }
= { 1, 2, 3, 4, 5, 6, 7, 8, 10, 12 }
Example.3) If, A = { a, b, c, d, e } and B = { w, x, y, z } , then find A U B = ?
Ans.) As per the given condition A = { a, b, c, d, e } and B = { w, x, y, z }
So, A U B = { a, b, c, d, e } U { w, x, y, z }
= { a, b, c, d, e, w, x, y, z }
Example.4) If A = { w, x, y, z }, then find A U A = ?
Ans.) As per the given condition A = { w, x, y, z }, we have to find out A U A
So, A U A = { w, x, y, z } U { w, x, y, z }
= { w, x, y, z }
= A
Then, A U A = A
The results we have arrived at are always true, in other words, they are laws of operations on sets.
Example.-5) If A = { a, b, c, d, e, f }, then find A U φ = ?
Ans.) As per the given condition A = { a, b, c, d, e, f }, we have to find out A U φ
Then, A U φ = { a, b, c, d, e, f } U φ
= { a, b, c, d, e, f }
= A
So, A U φ = A
The results we have arrived at are always true, in other words, they are laws of operations on sets.
Example.-6) If A = { a, b, c, d }, and B = { a, b, c, d, e, f, g, h } then find A U B
Ans.) As per the given condition A = { a, b, c, d } and B = { a, b, c, d, e, f, g, h }, we have to find out A U B.
So, A U B = { a, b, c, d } U { a, b, c, d, e, f, g, h }
Here we can observe that, A ⊆ B then
A U B = { a, b, c, d, e, f, g, h } = B
So, A U B = B, if A ⊆ B
The results we have arrived at are always true, in other words, they are laws of operations on sets.