Symbolizing with Identity

Multiple Choice: Pick the correct symbolization for each problem below. Use the following interpreation:

universe of discourse: coins in my pocket

Px: x is a penny

Qx: x is a quarter

Sx: x is silver

m: my half dollar

1. There is at least one quarter (in my pocket).

a.
(%x)Qx b.
(%x)Qx&(%x)~Qx c.
(^x)(Qxv~Qx)

2. There is at least one quarter but no
penny.

a.
(%x)Qx b.
(%x)Qx&~(%x)Px c.
(^x)(Qxv~Px)

3. There are at least two pennies.

a.
(%x)Px&(%y)Py&~Ixy b.
(%x)Qx&~(%x)Px c.
(^x)(Px>(Py&~Ixy)) d.
(%x)(%y)(~Ixy&(Px&Py))

4. There are exactly two quarters.

a.
(%x)(%y)(^z)[Qz=(IzxvIzy)] b.
(%x)(%y)(^z)[~Ixy&(Qz=(IzxvIzy))] c.
(%x)(%y)[~Ixy&(Qx&Qy)]

5. There are exactly two quarters. (For
the second time!)

a.
~(%x)(%y)(^z)[Qz=(IzxvIzy)] b.
(^x)(^y)(^z)[(Qx&Qy)&Qz)>((IxyvIxz)vIyz)] c.
(%x)(%y)[~Ixy&(Qx&Qy)]&(^x)(^y)(^z)[(Qx&Qy)&Qz)>((IxyvIxz)vIyz)]

6. The quarter in my pocket is made of
silver.

a.
(%x)(Qx&Sx) b.
~(%x)(%y)(~Ixy&(Qx&Qy)) c.
(%x)[(Qx&(^y)(Qy>Iyx)&Sx]

7. All coins in my pocket except my half-dollar
are pennies.

a.
(^x)(Cx>Ixm) b.
(^x)(~Ixm>Px) c.
(^x)(Ixm=Px)