Section 1: Problem 4 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

Let $A$ and $B$ be sets of real numbers. Write the negation of each of the following statements:

(a) For every $a\in A$ , it is true that $a^{2}\in B$ .

(b) For at least one $a\in A$ , it is true that $a^{2}\in B$ .

(d) For at least one $a\notin A$ , it is true that $a^{2}\in B$ .

Negation for (a): $\exists a\in A$ s.t. $a^{2}\notin B$ . (b) and (c) are negations of each other. (d): $\forall a\notin A:a^{2}\notin B$ .

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Section 1: Problem 4 Solution

Section 1: Problem 4 Solution