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hw1
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Homework assignment for MATH 4181 section 001, Fall 1999, from the Royster topology course folder. Seven problems, several cited by textbook exercise number, cover cardinality of finite Cartesian products, subsets that are not products, images and preimages under functions, injective and surjective cases, characteristic functions of sets, and the empty-set product.
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MATH 4181 001 Fall 1999
Problem Set 1
1. (Exercise 2, page 11) Prove that if Ahas precisely ndistinct elements and Bhas
preciselymdistinct elements, where mandnare positive integers, then ABhas
preciselymndistinct elements.
2. (Exercise 3, page 11) Let AandBbe sets, both of which have at least two distinct
members. Prove that there is a subset WABthat is not the product of a subset
ofAwith a subset of B.
3. (Exercise 1, page 14) Let f:A!Bbe given. Prove the following:
(a) For each subset XA,Xf 1(f(X)).
(b) For each subset YB,f(f 1(Y))Y.
(c) Iffis injective, then for each subset XA,
f 1(f(X)) =X:
(d) Iffis surjective, then for each subset YB,
f(f 1(Y)) =Y:
4. (Exercise 4, page 14) Let f:A!Bbe given.
(c) IfYis a subset of Bthenf 1(BnY) =An(f 1(Y)).
(d) IfXAandYB, then
f(X\f 1(Y)) =f(X)\Y:
5. (Exercise 7, page 15) Let Xbe a set and A;B;CX. The function A:X!f0;1g
de ned by
A(x) =(
1 ifx2A
0 ifx62A
is called the characteristic function of A. Show
(a) A\B= A B, where A B(x) = A(x) B(x).
(b) A[B= A+ B A\Band nd a similar expression for A[B[C.
6. Prove that;B=;for each set B.
7. Give an example to show that f(A\B) may not equal f(A)\f(B). Show that equality
holds iffis injective.
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