In operation of sets, we shall discuss about finding a new set by using fundamental mathematical operations on elements of sets.
Summary
Following are the various operations on sets:
- Intersection of Sets: Finding common elements between sets.
- Union of Sets: Finding total elements of given sets.
- Difference of Sets: Finding unique elements of a given set.
- Complement of Sets: Finding elements not contained in the given set.
- Symmetric difference: Finding set by difference of union of sets with their intersection.
Intersection of Sets
When we have two or more sets, a new set containing all the elements in common from the given sets is said to be the intersection of sets. It is represented by ‘∩‘ and referred as ‘CAP’.
SOLVED EXAMPLE 1
Q. Find the intersection between set A and set B where A = {1,2,3} and B = {2,3,4,5}.
Solution:
Given,
A = {1,2,3}
B = {2,3,4,5}
We know,
Intersection of sets A and B = A∩B
= {1, 2, 3} ∩ {2, 3, 4, 5}
= {2, 3}
From the above example, we can learn that the intersection of two sets is a new set that contains elements that are common in both the sets.
Union of Sets
When we have two or more sets, a new set containing all the elements without repetition from all the considered sets is said to be the union of sets. In other words, a Super Set formed by writing down all the elements present in given sets is said to be the union of sets. It is represented by ‘U‘ and referred as ‘CUP’.
SOLVED EXAMPLE 2
Q. Find the union between set A and set B where A = {1,2,3,4,5} and B = {4,5,6,7}.
Solution:
Given,
A = {1,2,3,4,5}
B = {4,5,6,7}
We know,
Union of sets A and B = A U B
= {1, 2, 3, 4, 5} U {4, 5, 6, 7}
= {1, 2, 3, 4, 5, 6, 7}
From the above example, we learn that the union of two sets is a new set that contains all the elements present in both sets, without repeating the common elements.
Difference of Sets
Difference of sets refers to a new set obtained by writing down the unique elements, excluding common elements, of a particular set, from the given sets. While finding the difference between two sets, we put a ‘-‘ (subtraction) symbol in between the names of sets. For example: A-B, B-C, A-C, etc.
We can also understand the difference of sets by a concept of only one set. When we subtract set B from set A, we mean only set A and not set B. So, we write down the elements of only set A. If the elements present in set A are also present in set B, we simply do not write them.
To bring light to what is mentioned above, let us see an example.
SOLVED EXAMPLE 3
Q. Find the difference of two sets A and B where A = {1,2,3,4,5} and B = {2,3,5}.
Solution:
Given,
A = {1,2,3,4,5}
B = {2,3,5}
We know,
Difference of sets A and B = A – B
= {1,2,3,4,5} – {2,3,5}
= {1,4}
[Here, we omitted 2,3 and 5 from set A which were the elements of set B]
Also,
Difference of sets B and A = B – A
= {2,3,5} – {1,2,3,4,5}
= {}
[Here, we omitted 2,3, and 5 from set B which were also the elements of set A and we are left with an empty set.]
From above example, we can list down the following things:
- Difference of two sets does not follow commutative property i.e. A – B ≠ B – A.
- The difference between set A and set B is a new set written by writing down the elements of set A by omitting the elements that are also present in set B.
- A – B = only A and B – A = only B.
Complement of Sets
Complement of a given set is the difference of that set from an Universal set. When we have a set A that is a proper subset of a universal set U, then a new set formed by writing down the elements present in the Universal U, excluding the common elements from set A, is said to be the complement of set A. Symbolically, complement of a set A is written as A’ or AC or $\overline{A}$.
Mathematically, A’ = AC = $\overline{A}$ = U – A
SOLVED EXAMPLE 4
Q. Find the complement of set A where A = {1,2,3,4,5} and U = {1,2,3,4,5,6,7,8}.
Solution:
Given,
A = {1,2,3,4,5}
B = {1,2,3,4,5,6,7,8}
We know,
Complement of set A (A’) = U – A
= {1,2,3,4,5,6,7,8} – {1,2,3,4,5}
= {6,7,8}
From above example, we can conclude the previously mentioned thing that: the complement of a set A is another set that doesn’t contain the elements of set A, also it is a subset of the Universal set. OR A’ ⊂ U.
We have repeated set A many times during the explanation of complement of sets. But, don’t get confused. You can write complement of other sets as:
- Complement of union of A and B = U – (A U B)
- Complement of intersection of A and B = U – (A ∩ B)
- Complement of set B = U – B
- Complement of set C = U – C and so on.
Symmetric difference
We have discussed about union, intersection, and difference of two sets at the beginning of this chapter. In symmetric difference of two sets, we can relate the ideas of all those three operations of sets.
First definition: Symmetric difference of two sets is the difference between union of two sets and intersection of the same sets.
Second definition: Symmetric difference of two sets is the union of differences of two sets (A-B) and (B-A) where A and B are two sets.
The symbol $\delta$ (delta) symbolizes the symmetric difference between two sets.
Mathematically,
(i) A △ B = (A U B) – (A ∩ B)
(ii) A △ B = (A – B) U (B – A)
SOLVED EXAMPLE 5
Q. Find the symmetric difference of set A and set B where A = {1,2,3,4,5} and B = {4,5,6,7,8}.
Solution:
Given,
A = {1,2,3,4,5}
B = {4,5,6,7,8}
Using formula (i)
A △ B = (A U B) – (A ∩ B)
= [{1,2,3,4,5} U {4,5,6,7,8}] – [{1,2,3,4,5} ∩ {4,5,6,7,8}]
= {1,2,3,4,5,6,7,8} – {4,5}
= {1,2,3,6,7,8}
Using formula (ii)
A △ B = (A – B) U (B – A)
= [{1,2,3,4,5} – {4,5,6,7,8}] U [{4,5,6,7,8} – {1,2,3,4,5}]
= {1,2,3} U {6,7,8}
= {1,2,3,6,7,8}
Whichever way we go, we get the same answer. However, formula (ii) A △ B = (A – B) U (B – A) is applied for finding symmetric difference of sets.
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