Relations and Functions (Class 12) · Question 4

Let the set of all relations R on the set \a, b, c, d, e, f\, such that R is reflexive and symmetric, and R contains exactly 10 elements, be denoted by S. Then the number of elements in S is .

Answer

105

Solution

Reflexive: all 6 pairs (a, a), , (f, f) must be in R, leaving 10 - 6 = 4 other ordered pairs.

Symmetric: off-diagonal pairs come in pairs \(x, y), (y, x)\, so we need exactly 2 such unordered pairs.

Number of unordered pairs of distinct elements = 62 = 15.

Number of relations = 152 = 105.

Answer: 105