Answer to Question #143627 in Abstract Algebra for Dolly

Question #143627
Let A={a,b,c} and P(A) the power set of A. List all the element of P(A). Show that the usual intersection, ∩ , and Union, U, of sets in P(A) are algebraic operation. What are the cayley's tables for there operation, Find the identity element if any, with respect to these operation.
1
Expert's answer
2020-11-11T19:34:58-0500

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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