These operations are well defined for any two subsets of A we obtain again a subset of A, as for any two subsets "B, C \\subset \\{a,b,c\\}, B\\cap C, B\\cup C \\subset \\{a,b,c\\}" .
Also we can see that these operations are commutative and associative (from the definition of an intersection and a union of sets). Therefore let us write explicitly the elements of P(A) :
"P(A) = \\{\\emptyset, \\{a\\}, \\{b\\}, \\{c\\}, \\{a,b\\}, \\{a,c\\}, \\{b,c\\}, \\{a,b,c\\}\\}"
And now the Cayley table are (by a direct calculation of union and intersection) :
We can clearly see either from definition of an intersection/union, either from the tables, that the identity element exists and the identity for "\\cup" is "\\emptyset" and the identity for "\\cap" is "\\{a,b,c\\}=A" .
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