First-order logic question: Which one is the translation of “John has exactly one brother ”?

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Can anyone help me walk through the following question?

I am struggling to interpret all the 4 options in this question...

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I tried to convert all the options into CNF, but that doesn't make more sense to me when I tried to do the interpretation.

first-order-logic logic
1 Answers

Translations:

(A) There exist x and y such that x is John's brother 
    and y is John's brother and x = y.
    <=> John has a brother.

(B) If there exists x such that x is John's brother then
    every y is John's brother and x = y.
    <=> If John has a brother then the brother is the
        only person in existence.

(C) If there exists x such that x is John's brother then
    for all y if y is John's brother then y = x.
    <=> If John has a brother then he only has one.

(D) If for all x, x is John's brother, then there exists
    a y such that y is John's brother and x = y.
    <=> If everybody is John's brother then John has
        a brother.

(E) There exists an x such that x is John's brother
    and for all y, if y is John's brother, then x = y.
    <=> John has a brother and he's the only one.

The option that corresponds to your statement is (E). It can't be the middle three because those are all conditional (and your statement that John has a brother is not conditional). (A) Does state that John has a brother; however, it does not say that brother is the only one (just that one exists). Technically, (A) says that John has a brother that is the same as the brother himself, which can be said of anybody who has at least one brother.

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