Section 2.1: Problem 3 Solution

Working problems is a crucial part of learning mathematics. No one can learn... 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

In 3–8, translate each English sentence into the first-order language specified. (You may want to carry out the translation in several steps, as in some of the examples.) Make full use of the notational conventions and abbreviations to make the end result as readable as possible.

Neither $a$ nor $b$ is a member of every set. ( $\forall$ , for all sets; $\in$ , is a member of; $a$ , $a$ ; $b$ , $b$ .)

English sentence “neither $a$ nor $b$ is $S$ ” means literally “ $a$ is not $S$ , and $b$ is not $S$ ”, hence, $\neg\forall xa\in x\wedge\neg\forall xb\in x$ , or, alternatively, $\exists xa\notin x\wedge\exists xb\notin x$ (but this latter sentence is a translation of a slightly different, though “logically equivalent”, English sentence “there is a set such that $a$ is not a member of it, and there is a set such that $b$ is not a member of it”).

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Section 2.1: Problem 3 Solution

Section 2.1: Problem 3 Solution