The relation \(\mathrm{R}\) defined on the let \(\mathrm{A}=\\{1,2,3,4,5\\}\) by \(R=\left\\{(x, y) /\left|x^{2}-y^{2}\right|<16\right\\} \quad\) is given by (a) \(\\{(1,1),(2,1),(3,1),(4,1),(2,3)\\}\) (b) \(\\{(2,2),(3,2),(4,2),(2,4)\\}\) (c) \(\\{(3,3),(4,3),(5,4),(3,4)\\}\) (d) None of these

Before testing the condition, let's write down all possible pairs in set A since it has only 5 elements. Remember that these relations are reflexive, so pairs like (1,1), (2,2), etc., should be considered. All possible pairs: {(1,1), (1,2), (1,3), (1,4), (1,5), (2,2), (2,3), (2,4), (2,5), (3,3), (3,4), (3,5), (4,4), (4,5), (5,5)}

Now let's apply the condition |x^2 - y^2| < 16 to each of the pairs and determine whether they satisfy the condition: (1,1): |1^2 - 1^2| = 0 < 16; true (1,2): |1^2 - 2^2| = |-3| = 3 < 16; true (1,3): |1^2 - 3^2| = |-8| = 8 < 16; true (1,4): |1^2 - 4^2| = |-15| = 15 < 16; true (1,5): |1^2 - 5^2| = |-24| = 24 < 16; false (2,2): |2^2 - 2^2| = 0 < 16; true (2,3): |2^2 - 3^2| = |-5| = 5 < 16; true (2,4): |2^2 - 4^2| = |-12| = 12 < 16; true (2,5): |2^2 - 5^2| = |-21| = 21 < 16; false (3,3): |3^2 - 3^2| = 0 < 16; true (3,4): |3^2 - 4^2| = |-7| = 7 < 16; true (3,5): |3^2 - 5^2| = |-16| = 16 < 16; false (4,4): |4^2 - 4^2| = 0 < 16; true (4,5): |4^2 - 5^2| = |-9| = 9 < 16; true (5,5): |5^2 - 5^2| = 0 < 16; true Valid pairs: {(1,1), (1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,3), (3,4), (4,4), (4,5), (5,5)}

Let's now compare the resultant set of valid pairs to the options given in the exercise: (a) {(1,1),(2,1),(3,1),(4,1),(2,3)} != our valid pairs (b) {(2,2),(3,2),(4,2),(2,4)} != our valid pairs (c) {(3,3),(4,3),(5,4),(3,4)} != our valid pairs The set of valid pairs we found does not match any of the given options, so the correct answer is: (d) None of these