1.11 Miscellaneous Examples

NCERT Class 11 Mathematics Textbook for blind and visually impaired students made screen readable by Professor T K Bansal.

Example 23

Show that the set of letters needed to spell “ CATARACT ” and the set of letters needed to spell “ TRACT” are equal.

Solution:

Let X be the set of letters in the word “CATARACT”. Then


X = { C, A, T, R }

Let Y be the set of letters in the word “ TRACT”. Then


Y = { T, R, A, C, T } = { T, R, A, C }

Since every element in X is in Y and every element in Y is in X. It follows that X = Y.

Example 24

List all the subsets of the set { −1, 0, 1 }.

Solution:

Let A = { −1, 0, 1 }.


The subset of A having no element is the empty set φ.


The subsets of A having only one element are { −1 }, { 0 }, { 1 }.


The subsets of A having two elements are {−1, 0}, {−1, 1} ,{0, 1}.


The subset of A having three elements of A is A itself.


So, all the subsets of A are


φ, {−1}, {0}, {1}, {−1, 0}, {−1, 1}, {0, 1} and {−1, 0, 1}.

Example 25

Show that A ∪ B = A ∩ B implies A = B

Solution:

Let a ∈ A. Then a ∈ A ∪ B. Since A ∪ B = A ∩ B , a ∈ A ∩ B. So a ∈ B.


Therefore, A ⊂ B. Similarly, if b ∈ B, then b ∈ A ∪ B. Since


A ∪ B = A ∩ B, b ∈ A ∩ B. So, b ∈ A. Therefore, B ⊂ A.


Thus, A = B