In class we saw that the monoid of the power set of a b c un

In class we saw that the monoid of the power set of {a, b, c} under Intersection is isomorphic to the set Z_2xZ_2xZ_2 under component-wise multiplication mod 2. Show that the power set of {a, b, c} under Intersection is also isomorphic to the power set of a 3 element set, say {d, e, f}, under Union. Explicitly show the bijection from one set to another, Show that the bijection maps the identity of one set to the identity of the other, and Give two examples of how the bijection preserves the binary operator.

Solution

power set of {a, b, c} is written as

{ {a},{b},{c},{a,b},{b,c}.{a,c}.{a,b,c},{phi} }

it is clearly isomorphic with power set of {d, e, f} as it can be written as,

{ {d},{e},{f},{d,e},{e,f}.{d,f}.{d,e,f},{phi} }