MAT 200 HOMEWORK 5
DUE IN CLASS ON 10/1
Problem 1. Let X be any set. For each A ∈ P(X), define the characteristic function χA : X →
{0, 1} by
(
1
if x ∈ A,
χA (x) =
0
if x ∈
/ A.
Let A, B ∈ P(X) and x ∈ X.
(i) Show that χA∩B (x) = χA (x)χB (x).
(ii) Give a formula for x 7→ χX\A (x) in terms of χA (x).
(iii) Using the fact that A \ B = A ∩ B c and B c = X \ B, give a formula for x 7→ χA\B (x).
(iv) Find the set C so that χC (x) = χA (x) + χB (x) − χA (x)χB (x).
Problem 2. Show that if f : X → Y is an injection and g : Y → Z is an injection, then g◦f : X → Z
is an injection.
Problem 3. Show that f : X → Y has a right inverse if and only if it is a surjection.
Problem 4. Let f : X → Y . Prove the following equalities hold for the induced functions
f : P(X) → P(Y ) and f −1 : P(Y ) → P(X).
(i) For all X1 , X2 ∈ P(X), f (X1 ∪ X2 ) = f (X1 ) ∪ f (X2 ).
(ii) For all Y1 , Y2 ∈ P(Y ), f −1 (Y1 ∩ Y2 ) = f −1 (Y1 ) ∩ f −1 (Y2 ).
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