## 1

(i) \$xWx

(ii)
\$x\$y[[WxÙWy]Ù¬x=y] (for example)

(iii)
\$x\$y\$z[[WxÙ[WyÙWz]]Ù[¬x=yÙ[¬y=zÙ¬z=x]]]

(iv)
¬\$x\$y[[WxÙWy]Ù¬x=y]; or, "x"y[[WxÙWy]®x=y]

(v)
¬\$x\$y\$z[[WxÙ[WyÙWz]]Ù[¬x=yÙ[¬y=zÙ¬z=x]]]; or "x"y"z[[WxÙ[WyÙWz]]®[x=yÚ[y=zÚz=x]]]
(vi) [\$xÙ¬\$x\$y[[WxÙWy]Ù¬x=y]]; or, [\$xÙ"x"y[[WxÙWy]®x=y]]; or \$x[WxÙ¬\$y[WyÙ¬x=y]]; or, \$x[WxÙ"y[Wy®x=y]]; or (but with a health warning!) \$x"y[Wy«x=y].

(vii) \$x[[WxÙLxa]Ù"y[[WyÙLya]®x=y]] (say)
(viii) \$x[[HxÙ[LxaÙLxb]] Ù ¬\$y[[HyÙ[LyaÙLyb]]Ù¬x=y]] (say)
(ix)
Answer. Best to take it slowly.
 First step: "x[Hx®x likes at most one woman who likes only one man] Next step: "x[Hx® there is at most one woman-who-likes-only-one-man who x likes] Next step: "x[Hx® ¬\$y\$z[[[y is a woman-who-likes-only-one-man Ù Lxy] Ù [z is a woman-who-likes-only-one-man Ù Lxz]] Ù ¬y=z]] Next step: y is a woman-who-likes-only-one-man becomes [WyÙ\$u[[MuÙLyu]Ù"v[[MvÙLyv]®u=v]]]; so we plug in this (and the formula that says the same thing about z) to give finally:
"x[Hx®¬\$y\$z[[[[WyÙ\$u[[MuÙLyu]Ù"v[[MvÙLyv]®u=v]]]ÙLxy]Ù [[WzÙ\$u[[MuÙLzu]Ù"v[[MvÙLzv]®u=v]]]ÙLxz]]Ù¬y=z]]

## 2

(i) There is at least one man. (Or just, there is a man.)
(ii) There is at least one man.
(iii) There are least two men.
(iv) There is at most one man.
(v) There is at most one man.
(vi) There are at least two men.
(vii) Exactly one woman likes John.
(viii) There is exactly one woman and she likes John.
(ix) There is a man.

(x) There is at most one man.  (It says that there is something to which any man is identical  - i.e. is one and the same thing as. So there can't be more than one man.)
(xi) Exactly one woman likes John.

(xii) There is a man. (It says that there is something such that, if anything is identical to it, it (the first thing) is a man. Since there is something identical to the first thing (namely it!), this means that the first thing is a man.)

(xiii) There is something, and, if it is man, it is the only thing there is.

(xiv) This means something rather peculiar. It is equivalent to "y\$x[¬x=yÚMx]. It is compatible with there being nothing at all (since it begins with """ and contains no names); it implies that, if there is just one thing, it is a man (since, if there is just one thing, there won't be anything which is not identical to it; so Mx will have to be true of it). But, if there is more than one thing, each could be a man, but none would have to be. So that is what it means: either there is nothing, or there is just one thing and it is a man, or there is more than one thing. (Not something one is likely to find oneself wanting to say!)

(xv) There is exactly one man whom every woman likes – i.e. the same man in each case.
(xvi) Every man likes just one woman (but not necessarily the same one).

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