1.

Using the following
interpretation:

Domain:
everything in 12 Main Street, Anytown


a:
John


b:
Mary


Mx:
x is a man


Wx:
x is a woman


Lxy:
x likes y


Hx:
x is human

formalise
the following:


(i)

There
is a least one woman 

(ii) 
There
are at least two women 

(iii)

There
are at least three women 

(iv)

The
is at most one woman 

(v)

There
are at most two women 

(vi)

There
is exactly one woman 

(vii) 
There
is exactly one woman who likes John 

(viii) 
Only
one person likes both John and Mary. 

(ix)

Everyone likes at most one woman who likes only one man. 
2.

Using
the same interpretation, translate the following into English: 

(i)

$xMx 

(ii)

$x$y[MxÙMy] 

(iii)

$x$y[[MxÙMy]Ù¬x=y] 

(iv)

"x"y[[MxÙMy]®x=y] 

(v)

¬$x$y[[MxÙMy]Ù¬x=y] 

(vi) 
¬"x"y[[MxÙMy]®x=y] 

(vii) 
$x[[WxÙLxa]Ù¬$y[[WyÙLya]Ù¬x=y]] 

(viii) 
$x[[WxÙLxa]Ù¬$y[WyÙ¬x=y]] 

(ix) 
$x"y[x=y®My] 

(x) 
$x"y[My®x=y] 

(xi) 
$x"y[[WyÙLya]«x=y] 

(xii) 
$x"y[x=y®Mx] 

(xiii) 
$x"y[Mx®x=y] 

(xiv) 
"y$x[x=y®Mx] 

(xv) 
$x[[Mx
Ù"y[Wy®Lyx]]Ù"z[[MzÙ"u[Wu®Luz]]®z=x] 

(xvi) 
"x[Mx®$y[[WyÙLxy]Ù"z[[WzÙLxz]®z=y]]] 