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ધારો કે $S =\{1,2,3,4\}$ તો ગણ $\{f: S \times S \rightarrow S : f$ એ વ્યાત્પ છે અને $f( a , b )=f( b , a ) \geqslant a ; \forall( a , b ) \in S \times S \}$ નાં ધટકોની સંખ્યા...........છે
$37$
$378$
$97$
$30$
Solution
$(1,1),(1,4),(4,1),(2,4),(4,2),(3,4),(4,3)$,
$(4,4)-$ all have one choice for image.
$(2,1),(1,2),(2,2)$ – all have three choices for image
$(3,2),(2,3),(3,1),(1,3),(3,3)-$ all have two choices for image.
So the total functions $=3 \times 3 \times 2 \times 2 \times 2=72$
Case $1$ : None of the pre-images have $3$ as image
Total functions $=2 \times 2 \times 1 \times 1 \times 1=4$
Case $2$ : None of the pre-images have $2$ as image
Total functions $=2 \times 2 \times 2 \times 2 \times 2=32$
Case $3$ : None of the pre-images have either $3$ or $2$
as image
Total functions $=1 \times 1 \times 1 \times 1 \times 1=1$
$\therefore$ Total onto functions $=72-4-32+1=37$
