Abstract Algebra: Theory and Applications · Problem 1.11

Preliminaries — Thomas Judson

Mathematics — Abstract Algebra · functions-mappings-inverse

Problem

Let

f : X \to Y

be a map with

A \subset X

and

B \subset Y

. ((a)) Prove

f(f^{-1}(B)) \subset B

. ((b)) Prove

f(f^{-1}(B)) = B

f

is onto. ((c)) Prove

A \subset f^{-1}(f(A))

. ((d)) Prove

f^{-1}(f(A)) = A

f

is one-to-one. ((e)) Either prove or give a counterexample to

f(A_1 \cap A_2) = f(A_1) \cap f(A_2)

for subsets

A_1

and

A_2

X

. ((f)) Prove

f(A_1 \cup A_2) = f(A_1) \cup f(A_2)

for subsets

A_1

and

A_2

X

. ((g)) Either prove or give a counterexample to

f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)

for subsets

B_1

and

B_2

Y

. ((h)) Prove

f^{-1}(B_1 \cup B_2) = f^{-1}(B_1) \cup f^{-1}(B_2)

for subsets

B_1

and

B_2

Y