Question
The number of relations on the set $\mathrm{A}=\{1,2,3\}$ containing at most 6 elements including (1, 2), which are reflexive and transitive but not symmetric, is __________.
✓
Answer
5
$A=\{1,2,3\}$
$(1,1),(2,2),(3,3),(1,2) \in R$
Remaining elements are
$(2,1),(2,3),(1,3),(3,1),(3,2)$
(1) If relation contains exactly 4 elements $=1$ way
(2) if relation contains exactly 5 elements
It can be $(1,3),(3,2) \Rightarrow 2$ ways
(3) If relation contain exactly 6 elements
It can be
$((2,3),(1,3)),((1,3),(3,2)),((3,1),(3,2))$
$\Rightarrow 3$ ways.
Total $=6$ ways
$A=\{1,2,3\}$
$(1,1),(2,2),(3,3),(1,2) \in R$
Remaining elements are
$(2,1),(2,3),(1,3),(3,1),(3,2)$
(1) If relation contains exactly 4 elements $=1$ way
(2) if relation contains exactly 5 elements
It can be $(1,3),(3,2) \Rightarrow 2$ ways
(3) If relation contain exactly 6 elements
It can be
$((2,3),(1,3)),((1,3),(3,2)),((3,1),(3,2))$
$\Rightarrow 3$ ways.
Total $=6$ ways
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