Quantifiers sample questions

Question: Some children lose every toy they own.

Answer Options:
a. (∃x)(Cx • (y)((Ty • Oxy) ⊃ Lxy)]
b. (x)((Cx • (y)(Ty • Oxy)) ⊃ Lxy)]
c. (x)((Cx ⊃ (y)(Ty • Oxy)) ⊃ Lxy)]
d. (∃x)(Cx ⊃ (y)(Ty • Oxy)) ⊃ Lxy]
e. (x)((Cx ⊃ (y)(Ty • Oxy)) ⊃ (y)Lxy]

Answer: a. (∃x)(Cx • (y)((Ty • Oxy) ⊃ Lxy)]

Question: The guests will be happy only if every room is clean and tidy.

Answer Options:
a. (∃x)(Gx ⊃ Hx) ⊃ (x)(Rx ⊃ (Cx • Tx)]
b. (x)(Rx ⊃ (Cx • Tx)) ⊃ (∃x)(Gx • Hx)
c. (∃x)(Gx • Hx) ⊃ (x)(Rx ⊃ (Cx • Tx))
d. (x)(Gx ⊃ Hx) ⊃ (Rx ⊃ (Cx • Tx))
e. (x)(Gx ⊃ Hx) ⊃ (x)(Rx ⊃ (Cx • Tx)]

Answer: e. (x)(Gx ⊃ Hx) ⊃ (x)(Rx ⊃ (Cx • Tx)]

Question: Miriam will be hired if and only if every manager approves.

Answer Options:
a. Hm ≡ (∃x)(Mx ⊃ Ax)
b. Hm ≡ (∃x)(Mx ⊃ Ax)
c. Hm ≡ (x)(Ax ⊃ Mx)
d. Hm ≡ (x)(Mx ⊃ Ax)
e. Hm ≡ (x)(Mx ⊃ Ax)

Answer: e. Hm ≡ (x)(Mx ⊃ Ax)

Question: Rollins is the shortest player on the team.

Answer Options:
a. Pr • (∃x)[(Px • x ≠ r) ⊃ Srx]
b. Pr • (x)[(Srx ⊃ x ≠ r)]
c. Pr • (∃x)[Px ⊃ Srx]
d. Pr • (x)[(Px ⊃ Srx)]
e. Pr • (x)[(Px • x ≠ r) ⊃ Srx]

Answer: e. Pr • (x)[(Px • x ≠ r) ⊃ Srx]

Question: Every city except Edenville was flooded.

Answer Options:
a. (x)[(Cx • x ≠ e) ⊃ Fx]
b. Ce • ∼Fe • (x)[(Cx • x ≠ e) ⊃ Fx]
c. (Ce • ∼Fe) ⊃ (x)[(Cx • x ≠ e) ⊃ Fx]
d. (x)[(Cx • Fe) ⊃ (Cx • x ≠ e) ⊃ Fx]
e. Ce • ∼Fe ⊃ (x)[(Cx ⊃ Fx) • x ≠ e]

Answer: b. Ce • ∼Fe • (x)[(Cx • x ≠ e) ⊃ Fx]

Question: Everyone fears someone (or other).

Answer Options:
a. (x)(y)(Px ⊃ Fxy)]
b. (x)(∃y)[(Px • Py) ⊃ Fxy)]
c. (x)(y)[(Px • Py) ⊃ Fxy)]
d. (x)(∃y)[(Px ⊃ Fxy)]
e. (∃x)(∃y)[(Px • Py) ⊃ Fxy]

Answer: d. (x)(∃y)[(Px ⊃ Fxy)]

Question: Megan is a biologist only if William is an Astronomer.

Answer Options:
a. Bm ⊃ Aw
b. Mb ⊃ Wa
c. Aw ⊃ Bm
d. (∃x)Bx ⊃ (∃x)Ax
e. (x)(Mx ⊃ Wx)

Answer: a. Bm ⊃ Aw

Question: A few dogs chase every cat they see.

Answer Options:
a. (∃x)(Dx ⊃ (y)((Cy ⊃ Sxy) ⊃ Cxy)]
b. (x)(Dx ⊃ (y)((Cy ⊃ Sxy) ⊃ Cxy)]
c. (∃x)(Dx ⊃ (y)((Cy ⊃ Sxy) ⊃ Cxy)]
d. (x)(Dx ⊃ (y)((Cy ⊃ Sxy) ⊃ Cxy)]
e. (∃x)(Dx ⊃ (y)((Cy ⊃ Sxy) ⊃ Cxy)]

Answer: e. (∃x)(Dx ⊃ (y)((Cy ⊃ Sxy) ⊃ Cxy)]

Question: A wallaby is a marsupial.

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

Answer: d. (∃x)(Wx ⊃ Mx)

Question: Angela wrote a poem.

Answer Options:
a. (∃x)(Px • Wax)
b. (x)(Wax ⊃ Px)
c. Wap
d. (∃x)(∃y)(Px ⊃ Wxy)
e. (∃x)(Px • Ax)

Answer: a. (∃x)(Px • Wax)

Question: Not all tennis players are high strung.

Answer Options:
a. (x)(Tx ⊃ ∼Hx)
b. (∃x)(Tx ⊃ Hx)
c. (∃x)(Tx ⊃ ∼Hx)
d. (x)(Hx ⊃ Tx)
e. (x)(Tx ⊃ ∼Hx)

Answer: c. (∃x)(Tx ⊃ ∼Hx)

Question: Every firefly glows in the dark.

Answer Options:
a. (x)(Fx ⊃ Gx)
b. (x)(Gx ⊃ Fx)
c. (∃x)(Fx ⊃ Gx)
d. (∃x)(Fx • Gx)
e. (∃x)(Hx ⊃ Fx)

Answer: a. (x)(Fx ⊃ Gx)