Consider the Cartesian product AB where AB are finite nonemp

Consider the Cartesian product A×B, where A,B are finite nonempty sets, each with cardinality greater than 1. There are two functions with domain A×B, called projections, with mapping rules p1(a,b) = a and p2(a,b) = b. What is the target space of p1? Of p2? Are either of p1, p2 one-to-one? Onto?

Solution

p1 projects elements of AxB into A

So target space of p1 is A

Similarly target space of p2 is B

None of them are one to one

BEcause let, r,s be two distinct element in B

p1(a,r)=p1(a,s)=a

Similarly

BEcause let, r,s be two distinct element in A

p2(r,b)=p2(s,b)=b

They are onto

Let, a be any element in A. Since B is non empty so thre ie some element in b in B and

p1(a,b)=a

So, p1 is onto

SImilarly p2 is onto