In this chapter we will learn some properties of set and then solve some practice questions.
The set properties are important to understand the concept of set better and get prepared for difficult set problems.
Properties of Set
Given below are some of the set properties with examples.
(01) In a set, the elements can be written in any order.
Hence, changing the order of elements will not affect the character of the given set.
For Example;
Consider the set A = { 4, 9, 13, 20 }
Here, the set name is A.
It has four elements 4, 9 , 13 and 20.
We can randomly change the order of elements and the set will remain the same.
A = { 13, 4, 9, 20 }
Note that we have reshuffled the elements in the set A.
Hence both the sets, A = { 4, 9, 13, 20 } and A = { 13, 4, 9, 20 } are the same.
Note:
While reshuffling the elements in a given set, you cannot introduce the new element in it, otherwise it would results in a new set.
For Example, consider the below set B.
B = { 11, 5, 13 }
Reshuffle the elements and insert new element 3 in it.
B’ = { 13, 11, 5, 3}
Here , \mathtt{B\ \neq \ B'} due to introduction of new element in the set B’
(02) In a set, the repeated elements can be removed so that only the distinct elements are left in the set.
Consider the below set A;
Note that elements 9 & 13 are present twice in the set A.
Removing the repeated digits, so that we have set with all the distinct elements.
Note that Set A = { 9, 13, 21, 13, 9 } is same as set A = {9, 13, 21 } because both are made of same type of digits 9, 13 & 21 which is repeated multiple times in the first set.
Example 02
X is a set made of letter in ” APPLE “
Note that in “APPLE”, the letter P is present two times.
In the set X, we can eliminate the repeated letters to make it distinct.
Hence, Set X = { A, P, L, E }
Example 03
Consider the set Y = { 1, 1, 2, 2, 3, 3, 4, 4, 5, 5 }
Removing the repeated digits in the set Y, we get;
Y = { 1, 2, 3, 4, 5 }
Note that Y = { 1, 1, 2, 2, 3, 3, 4, 4, 5, 5 } is same as Y = { 1, 2, 3, 4, 5 } as they are made of same element type 1, 2, 3, 4 & 5
(03) Two sets are equal when they have same distinct elements.
For Example, Consider the below sets;
A = { 3, -2, 7 }
B = { 7, 3, -2 }
Here, the set A is made of three elements 3, -2 & 7.
Set B is also made of same elements 3, -2, & 7.
Both the set is made of same three elements, it’s just that the position of elements are different.
As already discussed, the position of element do not affect the character of any set.
Hence, Set A = Set B.
Example 02
Consider the two sets below;
X = { R, A, B, B, I, T }
Y = { B, I, T, R, A }
Solution
In set X, the element B is repeated twice.
Eliminating the repeated elements we get the following set;
X = { R, A, B, I, T }
Hence, set x is made of 5 elements; R, A, B, I & T.
Now analyze set Y = { B, I, T, R, A }
Set Y is also made of 5 elements; R, A, B, I & T. It’s just that their position is different in the set.
Since, both set X & Y are made of same 5 elements, they are said to be equal set.
Properties of Set – Solved Example
(01) Simplify the below set by removing repeated elements
Solution
Note that the elements 25 and 75 are repeated twice in the given set.
On removing the repeated elements, we get;
A = { 25, 50 , 75 }
(02) Observe the below set and remove the repeated elements.
Solution
Observe that elements -6 and 16 are repeated twice.
Eliminating the repeated terms, we get the following set;
A = { -6, 8, 16, 9 }
(03) Check if the below expression is correct or not.
(a) { 2, -1, 3, 7 } = { 3, 2, -1, 7 }
(b) { 7, 10, -3} = { 10, -3, 8, 7 }
(c) { C, O, O, L } = { L, C, O }
(d) { P, O , T, A , T, O } = { T, O , P, A }
(e) { 6, 6, 6, 6 } = { 6 }
Solution
(a) { 2, -1, 3, 7 } = { 3, 2, -1, 7 }
The expression is correct.
The set { 2, -1, 3, 7 } is made of four elements 2, -1, 3 and 7.
The other set { 3, 2, -1, 7 } is also made of same elements 2, -1, 3 and 7. Only the position of elements have been shuffled here.
Hence, both the sets are equal to each other.
(b) { 7, 10, -3} = { 10, -3, 8, 7 }
Incorrect.
The first set contain three elements 7, 10 & -3.
The second set contain an addition element 8.
Hence, both the sets are not equal.
(c) { C, O, O, L } = { L, C, O }
Correct!!
The first set { C, O, O, L } contains three elements C, O & L.
The second set { L, C, O } also contains same element C, O, & L, it’s just that their position is shuffled.
(d) { P, O , T, A , T, O } = { T, O , P, A }
Correct!!
In set { P, O, T, A, T, O }, elements T & O are repeated twice.
Hence, the set { P, O, T, A, T, O } is made of four elements P, O, T, A.
The other set { T, O , P, A } is also made of four elements P, O, T, A.
Hence, both the sets are equal to each other.
(e) { 6, 6, 6, 6 } = { 6 }
Correct!!
In the above set, element 6 is repeated four times.
Removing the repeated elements, we get the set { 6 }
Hence, the above statement is correct.