The concept of operations on sets in mathematics is very close to the primary operations on numbers. In mathematics, we deal with the finite number of elements in sets. Every so often, a need arises to build up a relationship between two or more sets. We apply some operations on two or more sets to build up a relationship. These operations are called operations on sets in mathematics.
Preliminary to study about the types of different set operations, let us remember the concept of Venn Diagram because it would help you o know about the set operations well.
Venn Diagram:
A Venn diagram is a pictorial representation that represents the exact relationship between the finite collection of different sets.
Let us take a view of the Venn Diagram in the picture given below:
There are four different types of operations are the union of sets, the intersection of sets, the Difference of sets, and the complement of sets that we will study in this article.
Union of Sets:
The union of two sets in the operations of sets is also called the meeting of sets. We combine the elements of two sets to make a new set which is the union of two sets. The union of two sets is defined as:
“For any pair of sets, the union of two sets is a set which contains the elements that belong to the first set or the second set or both”.
In mathematics, a union of two sets S and T is the set that contains the elements of S and T or both. The union of two sets S and T is denoted by SUT.
To know the union of sets lets us take an example:
Example:
- Suppose that S={a, b, c, d} and T={f, g, h} then
SUT={a, b, c, d, f, g, h} which have the elements of both S and T.
- For E={2, 3, 5, 7} and F={1, 4, 6, 8}
EUF={1, 2, 3, 4, 5,6, 7,8}
Union of sets in Daily life
Union of sets with Venn Diagram:
The intersection of Sets:
The intersection of two sets in the operations of sets is also called the convergence of sets. We just take common elements from two sets to make a new net which is the intersection of two sets. It can be defined as:
“For pairs of sets, the intersection of two sets is a set that contains the common elements of both sets”.
In mathematics, the intersection of J and K is a set that contains common elements of J and K. The intersection of J and K is denoted by J∩K.
To understand the intersection of sets follows the examples given below:
Example:
- Let U={s, t, u, v} and V={u, v}
Then U∩V={u,v} which consists of common elements of both sets.
- For two sets L={±1, ±2, ±3} and M={-1, -2, -3}
Then L∩M={-1, -2, -3}
Example for the intersection of sets in Daily life:
The intersection of sets with Venn diagram:
Difference of Sets:
The difference of two sets in the operations of sets is obtained by subtracting the elements from two sets that are very close to the concept of the difference between the numbers. It can be defined as:
“The difference of two sets is a set that contains the elements of only the first set and not of second.”.
In mathematical form, The difference between two sets X and Y is a set that contains only the elements of X but not the elements of Y. The intersection of X and Y is denoted by X \ Y. Here the X \ Y also doesn’t contain common elements of X and Y.
Follow these examples to clarify the concept that is:
Example:
- If R={3, 6, 9, 12} and Q={4, 8, 12, 16}
Then R \ Q={3, 6, 9} which consists of the elements of R that are not in Q and also common elements between R and Q.
F={e, f, g, h} and G={a, b, e, f}
Then F \ G={g, h}
Examples for the Difference of sets in daily life:
Difference of sets with Venn Diagram:
Complement of Sets:
The complement of sets is an operation on sets that deals with universal sets.
“ A universal set is a set that contains the elements of all the sets that are under consideration”.
The complement of any set is a set that contains all the elements of the universal set which are not in the given set. In mathematics, it can be defined as:
“If S be any subsets of the universal set U then the complement of S is a set that contains all the elements of U which are not in S”.
The complement of any set is denoted by S’.Mathematically, it can be written as:
If S⊆U then S’ = U\S
Examples:
- If U={1, 2, 3, 4, 5, 6, 7, 8} and S={2, 3, 5, 7}
S’=U-S={1, 4, 6, 8} which contains elements of U which are not in S.
In daily life:
Practice Questions on Set Operations
Question#1: Deduce the union of the given sets.
(i) N = {2,4,6}, O = {3, 6, 9, 12}
(ii) I = {u, v, w}, J = {b, c, e}
(iii) X = {1, 4, 6, 8, 9}, Y = {5, 6, 7}
(iv) C = {i, o, u}, N = {1, 3, 5}
(v) G = {5, 10, 15, 20}, H = {6, 12, 18, 24,}
(vi) S = {a, b, o}, Q = {i, j, k}
Question#2: Work out the intersection of the following sets.
(i) U = {0, -1, -2, -3}, V = {–3, 2, -1, 0}
(ii) F = {2,4 ... , 10}, G = {2, 3, 5, 7}
(iii) R = {s, t, u, v, w}, S = {a, e, u, v, w}
(iv) U = {-1, -3, -5}, Z = {h, a, t}
(v) H = {a,s, l, m}, Y = {i, s, l, a, m},
(vi) V = {-2, -4, -6}, W = {0, ±1, ±2}
Question#3: If U = {1, 3, 5}, V = [2, 3, 4) and W = {3, 6, 9, 12}, then find:
(i) U \ W
(ii) V \ W
(iii) W \ U
(iv) U \ V
(v) W \ V
(vi) V \ U
Question#4 3. If U = {a, b, c,...., h }, P = {b, c, d, g}, R = {b, g}, and S = {a, e, h}, then find:
(i) P’
(ii) R’
(iii) S’
(iv) U’