Let R_1 and R_2 be relations on the set \ 1,2, ... ,50\ such that R_1 = \( p,p^n ):p is a prime and n >= 0 is an integer \ and R_2 = \( p,p^n ):p is a prime and n = 0 or 1\. Then, the number of elements in R_1 - R_2 is .

(8) : We have, R_1 = \( 2,2^0 ),( 2,2^1 ), ... ,( 2,2^5 ),( 3,3^0 ), ( 3,3^1 ),( 3,3^2 ),( 3,3^3 ),( 5,5^0 ),( 5,5^1 ),( 5,5^2 ),( 7,7^0 ),( 7,7^1 ),( 7,7^2 ), ( 11,11^0 ),(11,11),( 13,13^0 ),(13,13),( 17,17^0 ),(17,17),( 19,19^0 ), (19,19),( 23,23^0 ),(23,23),( 29,29^0 ),(29,29),( 31,31^0 ),(31,31), ( 37,37^0 ),(37,37),( 41,41^0 ),(41,41),( 43,43^0 ),(43,43),( 47,47^0 ), (47,47)\\ Similarly, R_2 = \( 2,2^0 )( 2,2^1 ),( 3,3^0 ),( 3,3^1 ), ... ,( 47,47^0 ), (47,47)\\ Therefore Number of elements in R_1 - R_2 = 8

Let R_{1} and R_{2} be relations on the set { 1,2,..... | Relation (Mathematics)

(8) : We have,

R_{1} = \{\left( 2,2^{0} \right),\left( 2,2^{1} \right),\ldots,\left( 2,2^{5} \right),\left( 3,3^{0} \right)

\left( 3,3^{1} \right),\left( 3,3^{2} \right),\left( 3,3^{3} \right),\left( 5,5^{0} \right),\left( 5,5^{1} \right),\left( 5,5^{2} \right),\left( 7,7^{0} \right),\left( 7,7^{1} \right),\left( 7,7^{2} \right)

\left( 11,11^{0} \right),(11,11),\left( 13,13^{0} \right),(13,13),\left( 17,17^{0} \right),(17,17),\left( 19,19^{0} \right)

(19,19),\left( 23,23^{0} \right),(23,23),\left( 29,29^{0} \right),(29,29),\left( 31,31^{0} \right),(31,31)

\left( 37,37^{0} \right),(37,37),\left( 41,41^{0} \right),(41,41),\left( 43,43^{0} \right),(43,43),\left( 47,47^{0} \right)

(47,47)\}

\\ Similarly,

R_{2} = \{\left( 2,2^{0} \right)\left( 2,2^{1} \right),\left( 3,3^{0} \right),\left( 3,3^{1} \right),\ldots,\left( 47,47^{0} \right)

(47,47)\}

\therefore\

Number of elements in

R_{1} - R_{2} = 8

Explanation

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