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)