## 1

(i)

There's no need to do it all in one step. We might proceed thus:

First step: ¬\$x[x is an undergraduate who worked less hard than John Ù Dx]

Next: ¬\$x[[UxÙHax]ÙDx]

(Altenatively: "x[[UxÙHax]®¬Dx])

(ii)

Definitely worth taking slowly.

First step: "x[x is an undergraduate who passed all their exams ® Dx]

Next: "x[[Ux Ù x passed all x's exams]®Dx]

Finally: "x[[UxÙ"y[Eyx®Pxy]]®Dx]

(iii)

First step: "x[x is an undergraduate who did not pass all their exams ® Hax]

Next step: "x[[UxÙ¬"y[Eyx®Pxy]]®Hax] (or, "x[[UxÙ\$y[EyxÙ¬Pxy]]®Hax]

(iv)

(First step: "x[Dx®x is an undergraduate who worked harder than John] and \$x[x is an undergraduate who worked harder than John Ù Dx] and \$x[x is an undergraduate who worked harder than John Ù ¬Dx]

Next step: [["x[Dx®[UxÙHxa]]Ù\$x[[UxÙHxa]ÙDx]]Ù\$x[[UxÙHxa]Ù¬Dx]]

(Or:  [[¬\$x[DxÙ¬[UxÙHxa]]Ù\$x[[UxÙHxa]ÙDx]]Ù\$x[[UxÙHxa]Ù¬Dx]])

Notice that, because none of the occurrences of "\$x" were within the scope of any other quantifier containing "x" we could stick to the same variable throughout.

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