Determine whether A = B, A ⊆ B, B ⊆ A, A ⊂ B, B ⊂ A or if none of these answers applies.
A = {x | x is a sport that uses a ball}
B = {basketball, soccer, tennis}
A = B No A doesn't equal B because for instance, A contains baseball, golf,
etc. but B doesn't.
A ⊆ B No, every element of A is not an element of B because for
instance, baseball is an element of A,
B ⊆ A Yes because, although B doesn't equal A by any means, every
element of B is also an element of A.
A ⊂ B No, this is just like A ⊆ B except that A ⊆ B allows
but does not require, the possibility of A = B, whereas A ⊂ B doesn't
allow A = B. But that's not any worry here.
B ⊂ A Yes because every element of B is also an element of A. The only
difference between ⊂ and ⊆ is that ⊂ does NOT allow the sets
on both sides of it to be exactly the same set, whereas ⊆ allows it but
does not require it.
So B ⊆ A and B ⊂ A are the only ones that hold.
Examples:
{a,r,t} = {t,a,r} is true because the order they're listed in doesn't matter
{a,r,t} ⊆ {t,a,r} is true because ⊆ allows (but doesn't require)
equality
{a,r,t} ⊂ {t,a,r} is false because ⊂ does not allow equality
{a,r,t} ⊂ {s,t,a,r} is true because every element of the left set is also
an element of the right set. It doesn't allow equality but we surely don't
have that.
{a,r,t} ⊆ {s,t,a,r} is true because every element of the left set is also
an element of the right set. It allows equality but doesn't require it.
Edwin