Section 5.07 Self-Quiz

. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)(Ax ⊃ Bx)
2. Aa
3. a=f(b)

Af(a) ⊃ Ba

Aa ⊃ Bf(a)

Aa ⊃ Bf(b)

Af(b)

Ab

. Which of the following propositions is derivable from the given premises in F?
1. (∀x)(Ax ⊃ Bx)
2. Aa
3. a=f(b)

(∃x)Bf(x)

(∀x)Ax

(∀x)Bf(x)

b=f(a)

Af(a)

. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)[Dx ⊃ Df(x)]
2. (∀x)[Ex ⊃ (Dx ≡ ~Fx)]
3. Ea • ~Fa

Ea ⊃ [Df(a) ≡ ~Ff(a)]

Ea ⊃ [Df(a) ≡ ~Fa]

Da ⊃ Da

Df(a) ⊃ Df(a)

Df(a) ⊃ Df(f(a))

. Which of the following propositions is derivable from the given premises in F?
1. (∀x)[Dx ⊃ Df(x)]
2. (∀x)[Ex ⊃ (Dx ≡ ~Fx)]
3. Ea • ~Fa

Df(f(a))

Ef(a)

Ef(f(a))

(∀x)Df(f(x))

Df(a)Ff(a)

. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)(∀y)f(x,y)=g(y,x)
2. e=f(a,b)
3. e=g(a,b)

(∀y)f(x,y)=g(y,a)

(∀y)f(a,y)=g(a,y)

(∀y)f(a,y)=g(y,a)

(∀y)f(g(a,b),y)=g(a,b)

(∀y)f(g(a,b),y)=g(b,g(b,a))

. Which of the following propositions is derivable from the given premises in F?
1. (∀x)(∀y)f(x,y)=g(y,x)
2. e=f(a,b)
3. e=g(a,b)

(∀x)(∀y)f(x,y)=f(y,x)

(∀x)(∀y)g(x,y)=g(y,x)

(∃x)(∃y)g(x,y)=g(y,x)

(∃x)f(x,f)=g(a,b)

f(b,a)≠f(a,b)

. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)(∀y)[f(x)=f(y) ≡ x=y]
2. Ja • ~Jb

(∀y)[x=y ≡ x=y]

(∀y)[a=f(y) ≡ a=y]

(∀y)[f(a)=f(y) ≡ a=y]

(∀y)[f(a)=f(y) ≡ f(a)=y]

(∀y)[f(a)=f(y) ≡ f(a)=f(y)]

. Which of the following propositions is derivable from the given premises in F?
1. (∀x)(∀y)[f(x)=f(y) ≡ x=y]
2. Ja • ~Jb

(∃x)(∃y)f(x)≠f(y)

(∃x)(∃y)f(x)=f(y)

a=b

(∀x)a=x

f(a)=f(b)

. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)[Lx ⊃ (~Mx ≡ Nx)]
2. (∃x)[Lf(x) • Mf(x)]

(∃x)[Lf(f(x)) • Mf(f(x))]

Lf(f(a)) • Mf(f(a))

Lf(e) • Mf(e)

La ⊃ (~Ma • Na)

Lf(a) ⊃ (~Ma ≡ Na)

. Which of the following propositions is derivable from the given premises in F?
1. (∀x)[Lx ⊃ (~Mx ≡ Nx)]
2. (∃x)[Lf(x) • Mf(x)]

(∀x)(Lx ⊃ ~Nx)

(∃x)(Lx • Nx)

(∃x)(Mx • Nx)

(∃x)Nx

(∃x)~Nx

. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)(∀y)(∀z)[f(x,y)=z ⊃ z=h(b)]
2. f(a,c)=b
3. f(a,b)=c

f(a,f(a,b))=c

f(b)=f(c)

(∀y)(∀z)[f(x,y)=z ⊃ z=b]

(∀y)(∀z)[f(a,y)=z ⊃ z=b]

(∀y)(∀z)[f(a,y)=z ⊃ z=h(b)]

. Which of the following propositions is derivable from the given premises in F?
1. (∀x)(∀y)(∀z)[f(x,y)=z ⊃ z=h(b)]
2. f(a,c)=b
3. f(a,b)=c

(∀x)(∀y)f(x,y)=f(y,x)

(∀x)(∀y)(∀z)[f(x,y)=z ⊃ f(x,z)=y]

f(b)=f(c)

h(b)=h(c)

b=c

. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x){Px ⊃ [Qf(x)x • Qxf(x)]}
2. Pa • a=f(b)

Pa ⊃ [Qf(a)a • Qaf(a)]

Pa ⊃ (Qaa • Qaa)

Pb ⊃ [Qf(a)b • Qbf(a)]

Pb

f(a)=b

. Which of the following propositions is derivable from the given premises in F?
1. (∀x){Px ⊃ [Qf(x)x • Qxf(x)]}
2. Pa • a=f(b)

Qbb

Qf(a)f(b) • Qf(b)f(a)

Qaf(b) • Qf(b)a

Qaa • Qbb

Qf(f(a))a • Qaf(f(a))

. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∀x)[Ax ⊃ x=f(j)]
2. (∀x)f(x)=g(x,a)
3. Ae • Af(e)

f(x)=g(a,a)

f(e)=g(e,a)

f(e)=g(f(e),a)

Ae ⊃ e=j

Aj ⊃ j=j

. Which of the following propositions is derivable from the given premises in F?
1. (∀x)[Ax ⊃ x=f(j)]
2. (∀x)f(x)=g(x,a)
3. Ae • Af(e)

(∃x)Ag(f(x),a)

(∃x)Ag(a,f(x))

Aj

e≠f(e)

f(e)≠g(f(e),a)

. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. Da • (∀x)[Dx ⊃ x=f(b)]
2. (∀x)[Ef(x) ≡ Ff(x)]
3. (∀x)(Dx ⊃ Ex)

Df(a)

Db

Da

Ea ≡ Fa

Ea ⊃ Da

. Which of the following propositions is derivable from the given premises in F?
1. Da • (∀x)[Dx ⊃ x=f(b)]
2. (∀x)[Ef(x) ≡ Ff(x)]
3. (∀x)(Dx ⊃ Ex)

(∃x)Fx

(∀x)(Ex ⊃ Fx)

Df(a)

(∃x)(Dx • Fx)

f(a)=f(b)

. Which of the following propositions is an immediate (one-step) consequence in F of the given premises?
1. (∃x)(∃y){Jx • Jy • (∀z)[Jz ⊃ (z=x ∨ z=y)] • Kxf(x) • Kyf(y)}
2. Ja • a=f(b)

Jb

Ja • b=f(a)

(∃y){Ja • Jy • (∀z)[Jz ⊃ (z=a ∨ z=y)] • Kaf(a) • Kyf(y)}

(∃y){Jm • Jy • (∀z)[Jz ⊃ (z=m ∨ z=y)] • Kmf(m) • Kyf(y)}

Ja • Jb • (∀z)[Jz ⊃ (z=a ∨ z=b)] • Kaf(a) • Kbf(b)

. Which of the following propositions is derivable from the given premises in F?
1. (∃x)(∃y){Jx • Jy • (∀z)[Jz ⊃ (z=x ∨ z=y)] • Kxf(x) • Kyf(y)}
2. Ja • a=f(b)

Kf(b)f(b)

Kf(b)f(f(b))

Kaa

(∃x)(∃y)(∃z)(Jx • Jy • Jz • x≠y • x≠z • y≠z)

(∃x)[Jx • (∀y)(Jy ⊃ y=x)]

Quiz Content