The number of relations on the set A = {1, 2, | vedclass

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(B) The set is $A = \{1, 2, 3\}$. For a relation $R$ to be reflexive,it must contain $(1, 1), (2, 2), (3, 3)$.Given $(1, 2) \in R$,for transitivity,if

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$6$

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(B) The set is $A = \{1, 2, 3\}$. For a relation $R$ to be reflexive,it must contain $(1, 1), (2, 2), (3, 3)$.Given $(1, 2) \in R$,for transitivity,if we add other elements,we must ensure the transitive property holds.Since $R$ must be reflexive and transitive but not symmetric,and $(1, 2) \in R$ but $(2, 1) 
otin R$ (to avoid symmetry),we analyze the possible relations with at most $6$ elements:$1$. If $R$ has $4$ elements: $R = \{(1, 1), (2, 2), (3, 3), (1, 2)\}$. This is reflexive and transitive,not symmetric. ($1$ way)$2$. If $R$ has $5$ elements: We add one element from ${(1, 3), (2, 3), (3, 1), (3, 2)}$. To maintain transitivity with $(1, 2)$,adding $(2, 3)$ gives $(1, 3)$,and adding $(3, 1)$ gives $(3, 2)$.Possible sets: ${(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}$ and ${(1, 1), (2, 2), (3, 3), (1, 2), (3, 1), (3, 2)}$. ($2$ ways)$3$. If $R$ has $6$ elements: We add two elements. Possible combinations that satisfy transitivity and reflexivity without symmetry are $3$ distinct sets.Total number of relations $= 1 + 2 + 3 = 6$.

The number of relations on the set $A = \{1, 2, 3\}$ containing at most $6$ elements including $(1, 2)$,which are reflexive and transitive but not symmetric,is . . . . . . .

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