Question

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.

Answer 1

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 $\{a,b,c\}=A$ .