Question

Find the cardinal number of the following sets:
(i)A = $\left\{ {x:x = {5^n},n \in {\mathbf{N}},n < 5} \right\}$
(ii)B = {x: x is a consonant in English Alphabets}
(iii)X = {x: x is a prime number}
(iv)P = {x: x<0, x$ \in $W}
(v)Q = {x: -3$ \leqslant x \leqslant $5, x$ \in $Z}

Answer 1

Hint: Cardinal number is a number that shows the amount of something present there. For e.g., one, two, three, etc. Here, we will first convert the sets into roster form from the set-builder form and then we will count the elements of the sets.

Complete step-by-step answer:
(i)A = $\left\{ {x:x = {5^n},n \in {\mathbf{N}},n < 5} \right\}$
For given set A, n can take values: 1, 2, 3 and 4 since n$ \in $N, n<5 i.e., A = {${5}^{1}$, ${5}^{2}$, ${5}^{3}$, ${5}^{4}$}
Therefore, the cardinal number of elements of set A is 4.

(ii)B = {x: x is a consonant in English Alphabets}
For a given set, x is a consonant in English alphabets. There are 26 alphabets in total and 5 (a, e, i, o, u) amongst them are termed as vowels. Hence, we are only left with 21 alphabets.
Therefore, consonants in English alphabets are 21.
 Hence, B = {b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}
So, we can say that the cardinal number of elements in set B is 21.

(iii)X = {x: x is a prime number}
The given set is an infinite set as prime numbers can’t be counted.
Therefore, the cardinal number will also be infinite for this set.

(iv)P = {x: x<0, x$ \in $W}
W represents the set of whole numbers.
In this set, x<0, x$ \in $W. It makes this set an infinite set as the whole number can’t be counted.
Therefore, the cardinal number will also be infinite for this set.

(v)Q = {x: -3$ \leqslant x \leqslant $5, x$ \in $Z}
For given set, the elements of the set will be: Q = {-3, -2, -1, 0, 1, 2, 3, 4, 5}
Therefore, the cardinal number of elements in the given set will be 9.

Note: In such questions, it is advisable to you that you should first learn with what all the sets are represented. By doing so, you will be able to relate with the question more quickly and you can find the roster form more easily. For e.g., the set of all real numbers is denoted by R.