Section 5.03 Self-Quiz

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. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)[Lx ⊃ (Oxy ≡ ~Oyx)]
2. (∀x)[(∃y)~Oyx ⊃ ~Mx]
3. (Pa • La) • Oab

La ⊃ (Oay ≡ ~Oya) correct incorrect

Pa correct incorrect

(∃y)~Oya ⊃ ~Mx correct incorrect

La ⊃ (Oya ≡ ~Oay) correct incorrect

Lb ⊃ (Oby ≡ ~Oyx) correct incorrect

* not completed

. Which of the following propositions is derivable from the given premises in F?
1. (∀x)[Lx ⊃ (Oxy ≡ ~Oyx)]
2. (∀x)[(∃y)~Oyx ⊃ ~Mx]
3. (Pa • La) • Oab

~Mb correct incorrect

Pa • ~Ma correct incorrect

La • Lb correct incorrect

Oba correct incorrect

~Oba • ~Mb correct incorrect

* not completed

. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x){[Mx • (∃y)Pxy] ⊃ Pxo}
2. (∀x)[(∃y)Qxy ≡ Pxa]
3. (∀x)(Mx ⊃ Nx)
4. (∀x)(Nx ⊃ Qxb)

Na ⊃ Qbb correct incorrect

(∃y)Qyy ≡ Pya correct incorrect

[Mo • (∃y)Poy] ⊃ Pxo correct incorrect

(∀x)(Qxo ≡ Pxa) correct incorrect

[Mx • (∃y)Pxy] ⊃ Pxo correct incorrect

* not completed

. Which of the following propositions is derivable from the given premises in F?
1. (∀x){[Mx • (∃y)Pxy] ⊃ Pxo}
2. (∀x)[(∃y)Qxy ≡ Pxa]
3. (∀x)(Mx ⊃ Nx)
4. (∀x)(Nx ⊃ Qxb)

(∀x)[Nx ⊃ (∃y)Pyx] correct incorrect

(∀x)[Nx ⊃ Pxo] correct incorrect

(∀x)[Mx ⊃ (∃y)Pxy] correct incorrect

(∀x)[(Mx • Nx) ⊃ ~(∀y)Pxy] correct incorrect

(∀x)[Mx ≡ Pxo] correct incorrect

* not completed

. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)[Ix ⊃ (∃y)Jyx]
2. (∀x)[Jxa ⊃ (Kx ∨ Lx)]
3. (∃x)Ix • (∀x)~Kx

Jxa ⊃ (Ka ∨ La) correct incorrect

(∃x)Ix correct incorrect

Ix ⊃ (∃y)Jxy correct incorrect

~Ka correct incorrect

Jaa ⊃ (Kx ∨ Lx) correct incorrect

* not completed

. Which of the following propositions is derivable from the given premises in F?
1. (∀x)[Ix ⊃ (∃y)Jyx]
2. (∀x)[Jxa ⊃ (Kx ∨ Lx)]
3. (∃x)Ix • (∀x)~Kx

(∀x)Lx correct incorrect

La correct incorrect

~Ka correct incorrect

(∃x)Lx correct incorrect

Jaa correct incorrect

* not completed

. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)(∀y)(Mxy ≡ ~Myx)
2. (∃x)(Kx • Mxa)
3. (∀x)(∀y)(~Mxy ⊃ Mxd)

~Mab ⊃ Mad correct incorrect

Mab ≡ ~Mba correct incorrect

~Mdd ⊃ Mdd correct incorrect

Kb • Mba correct incorrect

Kb • Maa correct incorrect

* not completed

. Which of the following propositions is derivable from the given premises in F?
1. (∀x)(∀y)(Mxy ≡ ~Myx)
2. (∃x)(Kx • Mxa)
3. (∀x)(∀y)(~Mxy ⊃ Mxd)

Mad correct incorrect

Mdd correct incorrect

Mbb correct incorrect

Mda correct incorrect

Maa correct incorrect

* not completed

. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)(Ax ⊃ ~Bx) ⊃ (∃x)Dx
2. (∀x)[Dx ⊃ (∃y)(Ey • Fxy)]
3. (∀x)[(∃y)Fyx ⊃ Gx]
4. ~(∃x)(Ax • Bx)

~(Aa • Ba) correct incorrect

(∃y)Fyy ⊃ Gy correct incorrect

Dx ⊃ (∃y)(Ey • Fay) correct incorrect

Ax ⊃ ~Bx correct incorrect

(∀x)~(Ax • Bx) correct incorrect

* not completed

. Which of the following propositions is derivable from the given premises in F?
1. (∀x)(Ax ⊃ ~Bx) ⊃ (∃x)Dx
2. (∀x)[Dx ⊃ (∃y)(Ey • Fxy)]
3. (∀x)[(∃y)Fyx ⊃ Gx]
4. ~(∃x)(Ax • Bx)

(∀x)(Ex ⊃ Gx) correct incorrect

(∃x)(Ex • Gx) correct incorrect

(∃x)(Dx • Gx) correct incorrect

(∀x)(Dx ≡ Gx) correct incorrect

(∃x)(Ex • Dx) correct incorrect

* not completed

. Consider assuming '(∀x)[Px ⊃ (∃y)(Qy • Rxy)]' for a conditional proof of the above logical truth. Which of the following propositions is a legitimate second step in that proof?
(∀x)[Px ⊃ (∃y)(Qy • Rxy)] ⊃ (∃x)(∃y)Rxy

Pa ⊃ (Qa • Raa) correct incorrect

Pa ⊃ (Qb • Rab) correct incorrect

Assume '~(∃x)(∃y)Rxy' for a nested indirect proof. correct incorrect

Assume '~(∃x)(∃y)Rxy' for a nested conditional proof. correct incorrect

(∃x)(∃y)Rxy correct incorrect

* not completed

. Which of the following propositions is also derivable in F?
(∀x)[Px ⊃ (∃y)(Qy • Rxy)] ⊃ (∃x)(∃y)Rxy

(∀x)[Px ⊃ (∃y)(Qy • Rxy)] ⊃ (∀x)(∀y)Ryx correct incorrect

(∀x)[Px ⊃ (∃y)(Qy • Rxy)] ⊃ (∃x)(∃y)Ryx correct incorrect

(∀x)[Px ⊃ (∃y)(Qy • Rxy)] ⊃ (∀x)(∀y)~Rxy correct incorrect

(∀x)[Px ⊃ (∃y)(Qy • Rxy)] ⊃ ~(∀x)(∀y)Rxy correct incorrect

(∀x)[Px ⊃ (∃y)(Qy • Rxy)] ⊃ ~(∀x)(∀y)~Rxy correct incorrect

* not completed

. Consider assuming '(∀x)(∀y)[(Px • Py) ⊃ Qxy]' for a conditional proof of the above logical truth. Which of the following propositions is a legitimate second step in that proof?
(∀x)(∀y)[(Px • Py) ⊃ Qxy] ⊃ (∃x)(∃y)Qxy

(∀y)[(Px • Py) ⊃ Qxy] correct incorrect

(∀x)[(Px • Py) ⊃ Qxy] correct incorrect

Assume '(∃x)(∃y)Qxy' for a nested indirect proof. correct incorrect

Px • Py correct incorrect

(Px • Py) ⊃ Qxy correct incorrect

* not completed

. Which of the following propositions is also derivable in F?
(∀x)(∀y)[(Px • Py) ⊃ Qxy] ⊃ (∃x)(∃y)Qxy

(∀x)(∀y)Qxy ⊃ (∃x)(∃y)[(Px • Py) • Qxy] correct incorrect

(∀x)(∀y)~Qxy ⊃ (∃x)(∃y)~[(Px • Py) • Qxy] correct incorrect

(∀x)(∀y)~Qxy ⊃ (∃x)(∃y)[(Px • Py) • ~Qxy] correct incorrect

(∀x)(∀y)Qxy ⊃ ~(∃x)(∃y)[(Px • Py) • Qxy] correct incorrect

(∀x)(∀y)Qxy ⊃ (∃x)(∃y)[(Px • Py) ⊃ ~Qxy] correct incorrect

* not completed

. Consider assuming '(∀x)[Px ⊃ (∀y)(Qy ⊃ Rxy)]' for a conditional proof of the above logical truth. Which of the following propositions is a legitimate second step in that proof?
(∀x)[Px ⊃ (∀y)(Qy ⊃ Rxy)] ⊃ [(∃x)(Px • Qx) ⊃ (∃x)Rxx]

Px correct incorrect

Assume '(∃x)(Px • Qx)' for a nested conditional proof. correct incorrect

Assume '~(∃x)Rxx' for a nested indirect proof. correct incorrect

Assume '(∃x)(Px • Qx) ⊃ (∃x)Rxx' for a nested indirect proof. correct incorrect

Pa ⊃ (Qa ⊃ Raa) correct incorrect

* not completed

. Which of the following propositions is also derivable in F?
(∀x)[Px ⊃ (∀y)(Qy ⊃ Rxy)] ⊃ [(∃x)(Px • Qx) ⊃ (∃x)Rxx]

(∀x)[(∀y)(Qy ⊃ Rxy) ⊃ Px] ⊃ [(∀x)(~Px ⊃ ~Qx) ∨ (∃x)Rxx] correct incorrect

(∀x)[Px ⊃ (∀y)(Qy ⊃ Rxy)] ⊃ [(∀x)(Px ⊃ ~Qx) ∨ (∃x)Rxx] correct incorrect

(∀x)[Px ⊃ ~(∃y)(Qy • Rxy)] ⊃ [(∀x)(Px ⊃ ~Qx) ∨ (∃x)Rxx] correct incorrect

(∀x)[Px ⊃ (∀y)(Qy ⊃ Rxy)] ⊃ [(∀x)(~Px ⊃ Qx) ∨ (∃x)Rxx] correct incorrect

(∀x)[Px ⊃ ~(∃y)(Qy • Rxy)] ⊃ [(∀x)(~Px ⊃ Qx) ∨ (∃x)Rxx] correct incorrect

* not completed

. Consider assuming '(∀x)(∀y)(Pxy ≡ Pyx)' for a conditional proof of the above logical truth. Which of the following propositions is a legitimate second step in that proof?
(∀x)(∀y)(Pxy ≡ Pyx) ⊃ [(∃x)Pax ⊃ (∃x)Pxa]

Assume '(∃x)Pax' for a nested conditional proof. correct incorrect

(∀x)(Pxy ≡ Pyx) correct incorrect

Pab ≡ Pba correct incorrect

Assume '~(∃x)Pxa' for a nested indirect proof. correct incorrect

Assume ~(∃x)Pax' for a nested indirect proof. correct incorrect

* not completed

. Which of the following propositions is also derivable in F?
(∀x)(∀y)(Pxy ≡ Pyx) ⊃ [(∃x)Pax ⊃ (∃x)Pxa]

(∀x)(∀y)(Pxy ≡ Pyx) ⊃ [(∀x)Pxa ⊃ (∀x)Pax] correct incorrect

(∀x)(∀y)(Pxy ≡ Pyx) ⊃ [(∃x)~Pxa ⊃ (∃x)~Pax] correct incorrect

[(∀x)~Pxa ⊃ (∀x)~Pax] ⊃ (∀x)(∀y)(Pxy ≡ Pyx) correct incorrect

(∀x)(∀y)(~Pxy ≡ Pyx) ⊃ [(∀x)Pax ⊃ (∀x)Pxa] correct incorrect

(∀x)(∀y)(Pxy ≡ Pyx) ⊃ [(∀x)~Pxa ⊃ (∀x)~Pax] correct incorrect

* not completed

. Which of the following is a good assumption for an indirect proof of the above logical truth?
~[(∀x)Pxa • (∀x)~Pbx]

(∃x)Pxa ∨ (∃x)Pbx correct incorrect

Pxa • Pbx correct incorrect

~(∀x)Pxa ∨ ~(∀x)~Pbx correct incorrect

~(∀x)Pxa correct incorrect

(∀x)Pxa • (∀x)~Pbx correct incorrect

* not completed

. Which of the following is a good assumption for a conditional proof of the above logical truth?
~[(∀x)Pxa • (∀x)~Pbx]

(∀x)Pxa correct incorrect

(∃x)Pbx correct incorrect

~(∃x)Pbx correct incorrect

~(∀x)Pxa correct incorrect

(∃x)(Pxa • Pbx) correct incorrect

* not completed

. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∃x)Bxa
2. (∃x)Cxa / (∃x)(Bxa • Cxa)

Valid correct incorrect

Invalid. Counterexample in a domain of two members, in which:
Baa: True     Caa: False
Bba: False     Cba: True correct incorrect

Invalid. Counterexample in a domain of two members, in which:
Baa: False     Caa: True
Bba: False     Cba: True correct incorrect

Invalid. Counterexample in a domain of two members, in which:
Baa: True     Caa: True
Bba: False     Cba: False correct incorrect

Invalid. Counterexample in a domain of two members, in which:
Baa: False    Caa: False
Bba: True     Cba: False correct incorrect

* not completed

. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∀x)[Ax ⊃ (∃y)(Ay • Fxy)]
2. Aa • ~Faa / (∀x)~Fxx

Valid correct incorrect

Invalid. Counterexample in a domain of one member, in which:
Aa: True     Faa: False correct incorrect

Invalid. Counterexample in a domain of two members, in which:
Aa: True     Faa: False     Fba: True
Ab: True     Fab: True     Fbb: True correct incorrect

Invalid. Counterexample in a domain of two members, in which:
Aa: True     Faa: False     Fba: False
Ab: False     Fab: True     Fbb: False correct incorrect

Invalid. Counterexample in a domain of two members, in which:
Aa: True     Faa: False     Fba: True
Ab: False     Fab: True     Fbb: True correct incorrect

* not completed

. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∀x)[Px ⊃ (∀y)(Ry ⊃ Txy)]
2. (∀x)[Qx ⊃ (∀y)(Ry ⊃ ~Txy)]
3. (∃x)Rx / ~(∃x)(Px • Qx)

Valid correct incorrect

Invalid. Counterexample in a domain of one member, in which:
Pa: False     Qa: True     Ra: True     Taa: False correct incorrect

Invalid. Counterexample in a domain of one member, in which:
Pa: True     Qa: False     Ra: True     Taa: True correct incorrect

Invalid. Counterexample in a domain of two members, in which:
Pa: True     Qa: True     Ra: True     Taa: False     Tba: True
Pb: False     Qb: True     Rb: True     Tab: False     Tbb: False correct incorrect

Invalid. Counterexample in a domain of two members, in which:
Pa: True     Qa: False     Ra: True     Taa: False     Tba: True
Pb: False     Qb: True     Rb: False     Tab: True     Tbb: False correct incorrect

* not completed

. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∃x)Faxb • (∃x)Fabx
2. (∀x)[(Faxb • Fabx) ⊃ Gx] / (∃x)Gx

Valid correct incorrect

Invalid. Counterexample in a domain of two members, in which:
Ga: False     Faab: True     Faba: False
Gb: False correct incorrect

Invalid. Counterexample in a domain of two members, in which:
Ga: False     Faab: False     Faba: True
Gb: False correct incorrect

Invalid. Counterexample in a domain of three members, in which:
Ga: False     Faab: True     Fabc: True
Gb: False     Faba: False     Facb: False
Gc: False     Fabb: False correct incorrect

Invalid. Counterexample in a domain of three members, in which:
Ga: False     Faab: False     Fabc: False
Gb: False     Faba: True     Facb: True
Gc: False     Fabb: True correct incorrect

* not completed

. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∀x){(Hx ⊃ (∃y)[Iy • (∀z)(Jz ⊃ Lxzy)]}
2. (∃x)(Hx • Jx)
3. (∀x)[(∃y)Lxxy ⊃ Kx] / (∃x)(Hx • Kx) • (∃x)(Jx • Kx)

Valid correct incorrect

Invalid. Counterexample in a domain of one member, in which:
Ha: True     Ja: True     Laaa: True
Ia: True     Ka: False correct incorrect

Invalid. Counterexample in a domain of one member, in which:
Ha: True     Ja: False     Laaa: False
Ia: False     Ka: True correct incorrect

Invalid. Counterexample in a domain of two members, in which:
Ha: True     Ja: True     Laaa: True     Lbaa: True
Hb: False     Jb: True     Laab: False     Lbab: True
Ia: True     Ka: True     Laba: True Lbba: False
Ib: True     Kb: False     Labb: True     Lbbb: False correct incorrect

Invalid. Counterexample in a domain of two members, in which:
Ha: True     Ja: False     Laaa: True     Lbaa: False
Hb: True     Jb: True     Laab: False     Lbab: True
Ia: True     Ka: True     Laba: False     Lbba: False
Ib: True     Kb: False     Labb: True     Lbbb: True correct incorrect