cartesian product and power set

[frctl]

Let A = {b, c, d, f, g}, B = {a, b, c}, and C = {d, f}.

What is the cartesian product set B x C?
{a,d, a,f}, {b,d b,f}, {c,d e,f}

What is the power set ℘(B)?
{a, b, c, ab, ac, bc, abc}

Are these correct?

 

[pka]

Let A = {b, c, d, f, g}, B = {a, b, c}, and C = {d, f}.
What is the cartesian product set B x C? {a,d, a,f}, {b,d b,f}, {c,d e,f} NO!
What is the power set ℘(B)?{a, b, c, ab, ac, bc, abc} & NO!
Are these correct?

A Cartesian product is a set of ordered pairs.
Example: \(\displaystyle \{u,v,w\}\times\{x,y\}=\{(u.x),(u.y),(v.x),(v.y),(w.x),(w.y\})\)

A power set is a set of subsets; example \(\displaystyle \mathscr{P}(\{a,b,c\})=\{\emptyset,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}\)

If \(\displaystyle \|A\|\) stands for the number of elements in a finite set then
the number of pairs in \(\displaystyle A\times B\) is \(\displaystyle \|A\|\cdot\|B\|\).
the number of sets in \(\displaystyle \mathscr{P}(A)\) is \(\displaystyle 2^{\|A\|}\)