Section 5.06 Self-Quiz

Select the best translation into predicate logic, using the following translation key:

a: true

b: false

p: Proposition P

q: Proposition Q

f(x): the truth-value of x

g(x): the truth value of ‘~x’

f(x,y): the truth value of ‘x ⊃ y’

g(x,y): the truth value of ‘x ≡ y’

Px: x is a proposition

. The truth-value of P is true and the truth value of Q is false.

f(p)=f(q)

f(p)≠g(q)

f(p)=a • g(q)≠a

f(p)=a • f(q)=b

f(p)=a ≡ f(q)=b

. P is true if, and only if, not-P is false.

f(p)=a • g(p)=a

f(p)=b • g(p)=a

f(p)=a ≡ g(p)=b

f(p)=a ≡ g(p)≠b

f(a)≠f(b)

. Every proposition is either true or false.

(∀x){Px ⊃ [f(x)=a ∨ f(x)=b]}

(∀x){Px ⊃ [f(x)=a ∨ g(x)=b]}

(∀x){Px ⊃ [f(x)=a ∨ b]}

(∀x){Px ⊃ [f(x)=g(x)]}

(∀x){Px ⊃ [Pf(a) • Pg(a)]}

. For any two propositions, if their conditional is false then the antecedent is true and the consequent is false.

(∀x)(∀y){(Px • Py) ⊃ {f(x,y)=b ⊃ [f(x)=b • f(y)=a]}}

(∀x)(∀y){(Px • Py) ⊃ {f(x,y)=b ⊃ [f(x)=a • f(y)=b]}}

(∀x)(∀y){(Px • Py) ⊃ {g(x,y)=b ⊃ [f(x)=a • f(y)=b]}}

(∀x)(∀y){(Px • Py) ⊃ {g(x,y)=a ⊃ [f(x)=a • f(y)=b]}}

(∀x)(∀y){(Px • Py) ⊃ {[f(x,y)=b • f(y,x)=b] ⊃ {[f(x)=b • f(y)=b] ∨ [f(x)=a • f(y)=b]}}}

. For any two propositions, if a biconditional between them is true then either they are both true or they are both false.

(∀x)(∀y){(Px • Py) ⊃ {f(x,y)=a ≡ [f(x)=f(y) ≡ f(y)=f(x)]}}

(∀x)(∀y){(Px • Py) ⊃ {f(x,y)=a ⊃ [f(x)=a • f(y)=a]}}

(∀x)(∀y){(Px • Py) ⊃ {g(x,y)=a ⊃ {[f(x)=a • f(y)=a] ∨ [f(x)=b • f(y)=b]}}}

(∀x)(∀y){(Px • Py) ⊃ {g(x,y)=a ⊃ {[f(x)=a • f(y)=a] ≡ [f(x)=b • f(y)=b]}}}

(∀x)(∀y){(Px • Py) ⊃ {g(x,y)=a ⊃ {[f(x)=a • g(x)=a] ∨ [f(x)=b • g(y)=b]}}}

. If a proposition is true, then any conditional in which it is the antecedent has the same truth value as the biconditional between it and the consequent of that conditional.

(∀x){Px ⊃ {f(x)=a ⊃ (∀y){Py ⊃ [f(y,x)=g(x,y)]}}}

(∀x){Px ⊃ {f(x)=a ⊃ (∀y){Py ⊃ [f(x,y)=g(x,y)]}}}

(∀x){Px ⊃ {f(x)=a ⊃ (∃y){Py • [f(y,x)=g(x,y)]}}}

(∀x){Px ⊃ {f(x)=a ⊃ (∃y){Py • [f(x,y)=g(x,y)]}}}

(∀x){Px ⊃ {f(x)=a ⊃ (∃y){[Py • f(x)=P] ⊃ [f(y,x)=a • g(x,y)=a]}}}

. If a proposition has a truth value (i.e. is either true or false), then so does its negation.

(∃x){Px • {[f(x)=a ∨ f(x)=b] ⊃ f(x)≠g(x)}}

(∃x){Px • {f(x)≠g(x) ⊃ [f(x)=a ∨ f(x)=b]}}

(∃x){Px • {[f(x)=a ∨ f(x)=b] ⊃ [g(x)=a ∨ g(x)=b]}}

(∀x){Px ⊃ {[f(x)=a ∨ g(x)=b] ⊃ f(x)=g(x)}}

(∀x){Px ⊃ {[f(x)=a ∨ f(x)=b] ⊃ [g(x)=a ∨ g(x)=b]}}

. No conditional with a true consequent is false.

(∀x){Px • [(∀y)f(y,x)=a ⊃ f(x)=b]}

(∀x){[Px • [(∀y)f(y,x)=b ⊃ f(x)≠b]}

(∀x){[Px • f(x)=a] ⊃ (∃y)[Py • f(y,x)≠b]}

(∀x){[Px • f(x)=a] ⊃ (∀y)[Py ⊃ f(y,x)≠b]}

(∀x){[Px • f(x)=a] ⊃ (∀y)[Py ⊃ f(x,y)≠b]

. Some true conditionals have true antecedents, while some have false antecedents.

(∃x)(∃y)[f(x,y)=a • f(x)=a] • (∃x)(∃y)[f(x,y)=a • f(x)=b]

(∃x)(∃y)[g(x,y)=a • g(x)=a] • (∃x)(∃y)[g(x,y)=a • g(x)=b]

f(m,n)=a • f(m)=a • f(m,n)=b • f(m)=b

f(m,n)=a • g(m)=a • f(m,n)=b • g(m)=b

(∃x)(∃y)[f(x,y)=a • f(x)=a] • (∃x)(∃y)[g(x,y)=a • g(x)=b]

. If two propositions have the same truth values, then the truth-values of either conditional made from those propositions is the same as the truth value of the biconditional made from those propositions.

f(p)=f(q) ⊃ [f(a,b)=g(a,b) • f(b,a)=g(a,b)]

f(p)=f(q) ⊃ [f(a,b)=g(a,b) • f(a,b)=g(b,a)]

(∀x)(∀y){[(Px • Py) • x=y] ⊃ [f(x,y)=g(x,y) • f(y,x)=g(x,y)]}

(∀x)(∀y){[(Px • Py) • f(x)=f(y)] ⊃ [f(x,y)=g(x,y) • f(y,x)=g(x,y)]}

(∀x)(∀y){[(Px • Py) • f(x,y)=f(y,x)] ⊃ [f(x,y)=g(x,y) • f(y,x)=g(x,y)]}

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