Let f S T Let U V S For any set U let fU fu u U 1 Prov

Let f : S ? T. Let U, V ? S. For any set U, let f(U) = {f(u) | u ? U}.

1. Prove that f(U ? V ) = f(U) ? f(V ).

2. Prove that f(U ? V ) ? f(U) ? f(V ) and that in general, f(U ? V ) 6= f(U) ? f(V )

Question 9 (15 points) Let f S T. Let U, V C S. For any set U, let f(U) f(u) E UY 1. Prove that f (UUV) f(U) U f(V) 2. Prove that f (Un V) C f(U) n f(V) and that in general, f(Un V) f(U) n f(V)

Solution

1. f(U V ) = f(U) f(V)

for example,

f(U)={1,3,5,7} ,f(V)={2,3,5,8} ,f(U V )=?

f(U V )={2,3,5,7,8}

f(U) f(V)={1,3,5,7}{2,3,5,8}

={2,3,5,7,8}

hence,

   f(U V ) = f(U) f(V) proved.

2.f(U V ) != f(U) f(V )

f(U V )= f(U) f(V)-f(U V )

hence,

  f(U V ) != f(U) f(V) proved