The number of relations, on the set {1,2,3} containing (1,2) and (2,3), which are reflexive and transitive but not symmetric, is __________.

Solution

Maths sir · 3 steps

Fix the pairs you are forced to include, then test the few leftover pairs one subset at a time.

1. Forced pairs. Reflexive forces (1,1), (2,2), (3,3). You are given (1,2) and (2,3), and transitivity then forces (1,3). So every valid relation contains these 6 pairs.
2. Pairs still free. Only (2,1), (3,2) and (3,1) are left, so there are 2³ = 8 choices. Each choice must stay transitive and must not make the relation symmetric.

   Extra pairs added	Transitive?	Not symmetric?	Counts?
   none	yes	yes, (2,1) missing	yes
   (2,1)	yes	yes, (3,2) missing	yes
   (3,2)	yes	yes, (2,1) missing	yes
   (3,1)	no: (3,1),(1,2) needs (3,2)	not needed, already fails	no
   (2,1), (3,2)	no: (3,2),(2,1) needs (3,1)	not needed, already fails	no
   (2,1), (3,1)	no: (3,1),(1,2) needs (3,2)	not needed, already fails	no
   (3,2), (3,1)	no: (2,3),(3,1) needs (2,1)	not needed, already fails	no
   all three	yes	no, this is the full relation	no

3. So. Exactly three relations pass both tests, so the answer is 3.

The common slip: 4. Counting the full relation A × A gives 4, but that relation is symmetric, so it is excluded.

The idea to keep. In counting questions on relations, first write down every pair the conditions force. Then test only the leftover pairs, checking each new pair against transitivity and the symmetry condition.

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