Solution : By Euclid’s Division Algorithm, we have a = bq + r …………..(i) On putting b = 3 in (1), we get a = 3q + r, [0 \(\le\) r < 3] If r = 0 a = 3q \(\implies\) \(a^2\) = \(9q^2\) …
Solution : To find the maximum number of columns, we have to find the H.C.F. of 616 and 32 i.e. 616 = 32 \(\times\) 19 + 8 and 32 = 8 \(\times\) 4 + 0 \(\therefore\) H.C.F of 616 and 32 is 8. Hence, maximum number of columns is 8.
Solution : By Euclid’s division algorithm, we have a = bq + r ……….(i) On putting, b = 6 in (1), we get a = 6q + r [0 \(\le\) r < 6] If r = 0, a = 6q, 6q is …
Question : Use Euclid’s division algorithm to find the H.C.F of : (i) 135 and 225 (ii) 196 and 38220 (iii) 865 and 225 Solution : (i) We start with the larger number 225. By Euclid’s Division Algorithm, we have 225 = 135 \(\times\) 1 + 90 We apply Euclid’s Division Algorithm on Division 135 …
Use Euclid’s division algorithm to find the H.C.F of : Read More »
Solution : We have, P(n, r) = \(n!\over (n – r)!\) Putting r = n, \(\implies\) P(n, n) = \(n!\over 0!\) \(\implies\) n! = \(n!\over 0!\) [ \(\because\) P(n, n) = n! ] \(\implies\) 0! = \(n!\over n!\) = 1 Hence, Proved.
Solution : If A and B are two sets having m and n elements respectively such that m \(\le\) n, then the total number of one-one functions from A to B is \(^nC_m \times m!\) where m! is m factorial. For example, Let set A have 3 elements and set B have 4 elements, then …
What is the formula to find number of one one function ? Read More »
Solution : Let A be a set. Then, the relation \(I_A\) = {(a, a) : a \(\in\) A} on A is called the identity relation on A. In other words, then the relation \(I_A\) on A is called the identity relation if every element of A is related to itself only. Example : If A …
Solution : Let A be a set. Then, A \(\times\) A \(\subseteq\) A \(\times\) A and so it is a relation on A. This relation is called the universal relation on A. In other words, a relation R on a set is called universal relation, if each element of A is related to every element …
Solution : Let A be a set. Then, \(\phi\) \(\subseteq\) A \(\times\) A and so it is a relation on A. This relation is called the void or empty relation on set A. In other words, a relation R on a set A is called void or empty relation, if no element of A is …
Solution : Let A be a finite set containing n elements. Let 0 \(\le\) r \(\le\) n. Consider those subsets of A that have r elements each. We know that the number of ways in which r elements can be chosen out of n elements is \(^nC_r\). Therefore, the number of subsets of A having …
Prove that the total number of subsets of a finite set containing n elements is \(2^n\). Read More »
