Symbolization questions

Question: The only daughter Robert has is Esther.

Answer Options:
a. (x)(Dxr ⊃ x = e)
b. Der • (x)(Dxr ⊃ x = e)
c. Der
d. Der • (x)(x = e ⊃ Der)
e. (∃x)(Dxr ⊃ x = e)

Answer: b. Der • (x)(Dxr ⊃ x = e)

Question: Elms and maples are deciduous trees.

Answer Options:
a. (x)[(Ex ∨ Mx) ⊃ (Dx ⊃ Tx)]
b. (x)[(Ex ∨ Mx) ⊃ (Tx • Dx)]
c. (x)[(Ex • Mx) ⊃ (Tx • Dx)]
d. (x)[(Ex ⊃ Mx) ⊃ (Tx • Dx)]
e. (∃x)[(Ex ∨ Mx) ⊃ (Tx • Dx)]

Answer: d. (x)[(Ex ⊃ Mx) ⊃ (Tx • Dx)]

Question: Ivan will be sad if and only if any child is injured.

Answer Options:
a. Si ≡ (∃x)(Cx ⊃ Ix)
b. (x)(Cx ⊃ Ix) ⊃ Si
c. (∃x)(Cx ⊃ Ix) ⊃ Si
d. Si ≡ (x)(Cx ⊃ Ix)
e. Si ≡ (Cx ⊃ Ix)

Answer: a. Si ≡ (∃x)(Cx ⊃ Ix)

Question: All the cakes and pies are delicious.

Answer Options:
a. (x)[Dx ⊃ (Cx ∨ Px)]
b. (x)[(Px • Cx) ⊃ Dx]
c. (x)[(Cx • Px) ⊃ Dx]
d. (x)[(Cx ∨ Px) ⊃ Dx]
e. (∃x)[(Px • Cx) ⊃ Dx]

Answer: d. (x)[(Cx ∨ Px) ⊃ Dx]

Question: If there are any guards, then if none of the prisoners escape, then they will be rewarded.

Answer Options:
a. (x)[(Gx ⊃ (y)(Py ⊃ ∼Ey)] ⊃ Rx)]
b. (x)[(Gx ⊃ (y)(Py ⊃ ∼Ey) ⊃ Rx)]
c. (x)[(Gx ⊃ (Py ⊃ ∼Ey)] ⊃ Rx)]
d. (x)[Gx ⊃ [(y)(Py ⊃ ∼Ey) ⊃ Rx)]
e. (x)[Gx ⊃ (Py ⊃ Ey) ⊃ Rx]

Answer: d. (x)[Gx ⊃ [(y)(Py ⊃ ∼Ey) ⊃ Rx)]

Question: A few scholarships were awarded.

Answer Options:
a. (∃x)(Sx ⊃ Ax)
b. (∃x)(Sx ⊃ Ax)
c. (x)(Sx ⊃ Ax)
d. Sy • Ay
e. (x)(Sx • Ax)

Answer: b. (∃x)(Sx ⊃ Ax)

Question: A freshman is not a sophomore.

Answer Options:
a. Fy ⊃ ∼Sy
b. (∃x)(Fx • Sx)
c. (∃x)(Fx ⊃ ∼Sx)
d. Fx • ∼Sx
e. (x)(Fx ⊃ ∼Sx)

Answer: e. (x)(Fx ⊃ ∼Sx)

Question: A taxi is waiting.

Answer Options:
a. (x)(Tx ⊃ Wx)
b. (∃x)(Tx ⊃ Wx)
c. (x)(Tx • Wx)
d. (∃x)(Tx • Wx)
e. Tx • Wx

Answer: c. (x)(Tx • Wx)

Question: There is at most one winner.

Answer Options:
a. (∃x)(∃y)(Wx • Wy)
b. (x)(y)((Wx • Wy) ⊃ x = y)]
c. (x)(y)((Wx • Wy) ⊃ x ≠ y]
d. (∃x)(y)((Wx • Wy) • x = y)]
e. (∃x)(y)((Wx • Wy) ⊃ e = y]

Answer: c. (x)(y)((Wx • Wy) ⊃ x ≠ y]

Question: A mouse is in the closet.

Answer Options:
a. (x)(Mx ⊃ Cx)
b. (∃x)(Mx ⊃ Cx)
c. (∃x)(Mx ⊃ Cx)
d. (x)(Mx • Cx)
e. (∃x)(Mx • Cx)

Answer: e. (∃x)(Mx • Cx)