The number of symmetric relations defined on the set {1,2,3,4} which are not reflexive is _________.

Solution

Maths sir · 4 steps

Count all symmetric relations first, then remove the ones that are also reflexive.

1. Split the pairs. The set has 4 elements, so A × A has 16 ordered pairs. There are 4 diagonal pairs like (1,1), and the other 12 group into 6 couples like {(1,2),(2,1)}.
2. Symmetric relations. Symmetry means each couple is taken together or left out together, which gives 2 choices per couple. Each diagonal pair can be in or out freely. That gives 2⁴ × 2⁶ = 2¹⁰ = 1024 symmetric relations.
3. Remove reflexive ones. A reflexive relation must contain all 4 diagonal pairs, so only the 6 couples are free. That gives 2⁶ = 64 relations that are both symmetric and reflexive.

   2¹⁰ - 2⁶ = 1024 - 64 = 960

4. So. The number of symmetric relations that are not reflexive is 960.

The common slip: 1024. Forgetting to subtract the 64 reflexive ones gives 1024, which counts every symmetric relation.

The idea to keep. On a set of n elements, there are 2^(n(n+1)/2) symmetric relations. Fixing the diagonal leaves 2^(n(n-1)/2) that are also reflexive. For 'not reflexive', subtract the second from the first.

Answer from the NTA final answer key. Solution by PrepWiser.