Section 4.08 Self Quiz

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. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∀x)(Ax ⊃ Bx)
2. (∃x)(Ax • ~Cx) / (∃x)(Bx • ~Cx)

Valid correct incorrect

Invalid. Counterexample in a domain of 1 member, in which:
Aa: True     Ba: False     Ca: False correct incorrect

Invalid. Counterexample in a domain of 1 member, in which:
Aa: True     Ba: False     Ca: True correct incorrect

Invalid. Counterexample in a domain of 1 member, in which:
Aa: False     Ba: False     Ca: True correct incorrect

Invalid. Counterexample in a domain of 1 member, in which:
Aa: False     Ba: False     Ca: False correct incorrect

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. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∀x)(Hx ⊃ Ix)
2. (∃x)(Hx • ~Jx)
3. Ja / (∃x)(Jx • ~Hx)

Valid correct incorrect

Invalid. Counterexample in a domain of 1 member, in which:
Ha: True     Ia: True     Ja: False correct incorrect

Invalid. Counterexample in a domain of 1 member, in which:
Ha: True     Ia: True     Ja: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Ha: False     Ia: False     Ja: True
Hb: True     Ib: True     Jb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Ha: True     Ia: True Ja: True
Hb: True     Ib: True     Jb: False correct incorrect

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

Valid correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Ha: False     Ia: False Ga: True
Hb: True     Ib: True     Gb: False correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Ha: True     Ia: False Ga: True
Hb: True     Ib: True     Gb: False correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Ha: True     Ia: False Ga: True
Hb: False     Ib: True     Gb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Ha: False     Ia: False Ga: True
Hb: False     Ib: True     Gb: True correct incorrect

* not completed

. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∀x)[Dx ⊃ (Ex ∨ Fx)]
2. (∀x)(Dx ⊃ ~Ex) / (∀x)(Dx ⊃ Fx)

Valid correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Da: False     Ea: False Fa: True
Db: False     Eb: True     Fb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Da: False     Ea: False Fa: True
Db: False     Eb: False     Fb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Da: False     Ea: True Fa: True
Db: False     Eb: True     Fb: False correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Da: False     Ea: False Fa: True
Db: True     Eb: False     Fb: True correct incorrect

* not completed

. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∀x)(Fx ⊃ Gx)
2. (∀x)(Gx ⊃ ~Hx)
3. (∃x)Fx / (∃x)(Fx • ~Hx)

Valid correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Fa: False     Ga: False Ha: False
Fb: True     Gb: False Hb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Fa: False     Ga: False     Ha: True
Fb: False     Gb: False Hb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Fa: False     Ga: False     Ha: True
Fb: True     Gb: True Hb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Fa: False     Ga: True     Ha: True
Fb: True     Gb: False Hb: False correct incorrect

* not completed

. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∀x)(Ax ⊃ Bx)
2. (∀x)(Dx ⊃ Bx) / (∃x)(Ax • Dx)

Valid correct incorrect

Invalid. Counterexample in a domain of 1 member, in which:
Aa: False     Ba: False     Da: False correct incorrect

Invalid. Counterexample in a domain of 1 member, in which:
Aa: True     Ba: False     Da: False correct incorrect

Invalid. Counterexample in a domain of 1 member, in which:
Aa: True     Ba: False     Da: True correct incorrect

Invalid. Counterexample in a domain of 1 member, in which:
Aa: False     Ba: False     Da: True correct incorrect

* not completed

. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∀x)(Nx ⊃ ~Ox)
2. (∀x)(Px ⊃ Ox) / ~(∃x)(Nx • Px)

Valid correct incorrect

Invalid. Counterexample in a domain of 1 member, in which:
Na: False     Oa: False     Pa: False correct incorrect

Invalid. Counterexample in a domain of 1 member, in which:
Na: True     Oa: False     Pa: False correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Na: True     Oa: False     Pa: False     Nb: True     Ob: True     Pb: False correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Na: False     Oa: False     Pa: False
Nb: False     Ob: False     Pb: False correct incorrect

* not completed

. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∀x)(Kx ⊃ ~Lx)
2. (∃x)(Mx • Lx) / (∀x)(Kx ⊃ ~Mx)

Valid correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Ka: False     La: False     Ma: True
Kb: False     Lb: True     Mb: False correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Ka: True     La: True     Ma: True
Kb: False     Lb: False     Mb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Ka: True     La: False     Ma: True
Kb: False     Lb: True     Mb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Ka: False     La: False     Ma: True
Kb: False     Lb: False     Mb: False correct incorrect

* not completed

. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∃x)(Ax • Bx)
2. (∀x)(Bx ⊃ Cx)
3. (∀x)(Cx ⊃ ~Dx)
4. (∀x)(Ex ⊃ Dx) / (∃x)(Ax • ~Ex)

Valid correct incorrect

Invalid. Counterexample in a domain of 1 member, in which:
Aa: True     Ba: False     Ca: True     Da: True     Ea: True correct incorrect

Invalid. Counterexample in a domain of 1 member, in which:
Aa: True     Ba: True     Ca: True     Da: False     Ea: False correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Aa: True     Ba: True     Ca: True     Da: True     Ea: True
Ab: True     Bb: False     Cb: True     Db: True     Eb: False correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Aa: True     Ba: True     Ca: True     Da: True     Ea: True
Ab: False     Bb: False     Cb: True     Db: False     Eb: False correct incorrect

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

Valid correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Pa: True     Qa: True     Ra: False     Sa: True
Pb: True     Qb: True     Rb: False     Sb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Pa: True     Qa: True     Ra: True     Sa: False
Pb: False     Qb: False     Rb: False     Sb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Pa: False     Qa: False     Ra: False     Sa: False
Pb: False     Qb: False     Rb: True     Sb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Pa: True     Qa: True     Ra: True     Sa: False
Pb: True     Qb: False     Rb: False     Sb: False correct incorrect

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. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. Pa • Qa
2. (∃x)(Px • Rx)
3. (∀x)(Qx ⊃ ~Rx) / (∀x)(Sx ⊃ Px)

Valid correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Pa: True     Qa: True     Ra: False     Sa: False
Pb: True     Qb: False     Rb: True     Sb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Pa: True     Qa: True     Ra: False     Sa: True
Pb: True     Qb: False     Rb: True     Sb: False correct incorrect

Invalid. Counterexample in a domain of 3 members, in which:
Pa: True     Qa: True     Ra: False     Sa: False
Pb: True     Qb: False     Rb: True     Sb: False correct incorrect

Pc: False     Qc: False     Rc: True     Sc: True correct incorrect

Invalid. Counterexample in a domain of 3 members, in which:
Pa: True     Qa: True     Ra: False     Sa: True
Pb: True     Qb: False     Rb: True     Sb: False
Pc: False     Qc: True     Rc: True     Sc: False correct incorrect

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. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. Ed • Fd
2. (∀x)(Ex ⊃ Gx)
3. (∀x)(Fx ⊃ Hx) / ~(∀x)(Gx ⊃ ~Hx)

Valid correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Ed: True     Fd: True     Gd: True     Hd: True
Ed: True     Fd: True     Gd: False     Hd: False correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Ed: True     Fd: True     Gd: True     Hd: True
Ed: False     Fd: True     Gd: False     Hd: False correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Ed: True     Fd: True     Gd: True     Hd: True
Ed: True     Fd: False     Gd: False     Hd: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Ed: True     Fd: True     Gd: True     Hd: True
Ed: False     d: True     Gd: True     Hd: False correct incorrect

* not completed

. Determine whether the given argument is valid or invalid. If it is invalid, select a counterexample.
1. (∀x)(Px ⊃ Qx)
2. (∃x)(Px • Rx)
3. Ra / Ra • Qa

Valid correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Pa: True     Qa: False     Ra: False
Pb: True     Qb: True     Rb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Pa: False     Qa: False     Ra: True
Pb: True     Qb: False     Rb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Pa: False     Qa: False     Ra: True
Pb: True     Qb: True     Rb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Pa: False     Qa: False     Ra: True
Pb: False     Qb: True     Rb: True correct incorrect

* not completed

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

Valid correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Pa: True     Qa: False     Ra: False
Pb: True     Qb: True     Rb: True correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Pa: True     Qa: False     Ra: False
Pb: False     Qb: False     Rb: False correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Pa: True     Qa: False     Ra: False
Pb: False     Qb: True     Rb: False correct incorrect

Invalid. Counterexample in a domain of 2 members, in which:
Pa: True     Qa: False     Ra: True
Pb: True     Qb: True     Rb: False correct incorrect

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. Determine whether the given proposition is a logical truth of M or not. If it is not a logical truth, select a false valuation.
~(∀x)(Bx ⊃ ~Cx) ≡ (∃x)(Bx • Cx)

Logical truth correct incorrect

Not a logical truth. False valuation in a domain of 1 member, in which:
Ba: True     Ca: False correct incorrect

Not a logical truth. False valuation in a domain of 1 member, in which:
Ba: True     Ca: True correct incorrect

Not a logical truth. False valuation in a domain of 2 members, in which:
Ba: True     Ca: True
Bb: False     Cb: True correct incorrect

Not a logical truth. False valuation in a domain of 2 members, in which:
Ba: True     Ca: True
Bb: True     Cb: False correct incorrect

* not completed

. Determine whether the given proposition is a logical truth of M or not. If it is not a logical truth, select a false valuation.
(∀x)[Dx ⊃ (Ex ∨ Fx)] ⊃ (∀x)(Dx ⊃ Ex)

Logical truth correct incorrect

Not a logical truth. False valuation in a domain of 1 member, in which:
Da: False     Ea: True     Fa: True correct incorrect

Not a logical truth. False valuation in a domain of 1 member, in which:
Da: True     Ea: False     Fa: True correct incorrect

Not a logical truth. False valuation in a domain of 1 member, in which:
Da: True     Ea: False     Fa: False correct incorrect

Not a logical truth. False valuation in a domain of 1 member, in which:
Da: False     Ea: False     Fa: True correct incorrect

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. Determine whether the given proposition is a logical truth of M or not. If it is not a logical truth, select a false valuation.
(∃x)[Gx • (Hx ∨ Jx)] ∨ ~(∃x)(Gx • Hx)

Logical truth correct incorrect

Not a logical truth. False valuation in a domain of 1 member, in which:
Ga: True     Ha: False     Ja: False correct incorrect

Not a logical truth. False valuation in a domain of 1 member, in which:
Ga: False     Ha: False     Ja: False correct incorrect

Not a logical truth. False valuation in a domain of 1 member, in which:
Ga: True     Ha: True     Ja: True correct incorrect

Not a logical truth. False valuation in a domain of 1 member, in which:
Ga: False     Ha: True     Ja: True correct incorrect

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. Determine whether the given proposition is a logical truth of M or not. If it is not a logical truth, select a false valuation.
(∃x)[(Lx • Mx) • Nx] ⊃ ~(∀x)[~(Lx • Mx) ⊃ Nx]

Logical truth correct incorrect

Not a logical truth. False valuation in a domain of 1 member, in which:
La: True     Ma: True     Na: True correct incorrect

Not a logical truth. False valuation in a domain of 1 member, in which:
La: False     Ma: True     Na: True correct incorrect

Not a logical truth. False valuation in a domain of 1 member, in which:
La: True     Ma: False     Na: True correct incorrect

Not a logical truth. False valuation in a domain of 1 member, in which:
La: True     Ma: True     Na: False correct incorrect