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# Defining Certain Subsets of the Natural Numbers

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Let L be the language of addition (with equality) in first-order logic. That is, let L be the first-order language that allows for use of the equality symbol ("=") and whose only non-logical symbol is a binary function symbol "+". (That is, L has no constant symbols and no predicate (relation) symbols.)

Now consider the L-structure that has as universe the set N of all the natural numbers (0, 1, 2, ...) and where the function symbol "+" is interpreted as the usual addition on the natural numbers.

(a) Use an L-formula of first-order logic to define the set E_1 of all the even natural numbers.

(b) Use an L-formula of first-order logic to define the set E_2 of all the even natural numbers that are not divisible by 4.

(c) Use an L-formula of first-order logic to define the set S={1} (i.e., S is the one-element subset of the natural numbers whose only element is the number 1).