Section 1: Problem 7 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Given sets $A$ , $B$ , and $C$ , express each of the following sets in terms of $A$ , $B$ , and $C$ , using the symbols $\cup$ , $\cap$ , and $-$ .

$D=\{x|x\in A\mbox{ and }(x\in B\mbox{ or }x\in C)\},$ $E=\{x|(x\in A\mbox{ and }x\in B)\mbox{ or }c\in C\},$ $F=\{x|x\in A\mbox{ and }(x\in B\Rightarrow x\inС)\}.$

$D=A\cap(B\cup C)$ , $E=(A\cap B)\cup C$ , $F=A-(B-C)$ . In the last set, $\{x|x\in B\Rightarrow x\in C\}$ is the set of all points such that if $x$ happens to be in $B$ then it must also be in $C$ , but if $x\notin B$ , then $x$ can be any point. In other words, the set is the set of all points except those that are in $B-C$ . Hence, the intersection of this set with $A$ is the set of all points in $A$ that are not in $B-C$ .

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Section 1: Problem 7 Solution

Section 1: Problem 7 Solution