MATH 160, MATHEMATICAL LOGIC Midterm Review ... - Mathematics

MATH 160, MATHEMATICAL LOGIC Midterm Review questions, Spring 2007.
1. Either prove by using truth tables or disprove by giving a counter example (a particular
truth assignment where the implication fails)
(a) α −→ β |= β −→ α
(b) (α ∨ β) ∧ γ |= α ∨ (β ∧ γ)
2. Let φ be the wff (α −→ β) −→ γ
(a) Find a wff tautologically equivalent to φ which uses only the connectives ¬ and ∧.
(b) Find another wff tautologically equivalent to φ which is in disjunctive normal form.
3. Let ⊕ denote exclusive or.
(a) Is α ∧ ¬α (a contradiction) expressible with just {∨, ∧, ⊕}. If so, how ? If not,
explain why ?
(b) Is {∨, ∧, ⊕} complete? Prove your answer. Your argument needs to be convincing,
but obvious facts do not need detailed proofs.
4. Give a proof from the axioms of propositional calculus of the formula ¬(α −→ ¬β) −→
α
5. Let P C denote the theory of propositional calculus (= set of wff’s provable from axiom
schemes L 1 , L 2 , L 3 using Modus Ponens) Let P C + be the extension of P C obtained by
adding a fourth axiom scheme
(¬α −→ β) −→ (α −→ ¬β)
(a) Show that P C is a proper subset of P C + .
(b) We say a theory is consistent if there is a wff α such that α is not provable. Is
P C + consistent? Prove your answer. Hint: you might try γ −→ γ in place of α and β
in the scheme.
6. Let the formula σ i be defined as follows:
Now define
σ i = (P 1 ∨ P 2 ∨ . . . ∨ P i ) −→ P i+1 , i ≥ 1
Σ = {σ i : i = 1, 2, 3, . . .}
Describe all truth assignments ν that satisfy Σ.
7. Let P C ′ be the theory in the language of propositional calculus which differs from P C
only in having the axiom scheme
L ′ 3 : (¬α −→ ¬β) −→ ((¬α −→ β) −→ α))
in place of L 3 . Show that P C = P C ′ 1

8. Assume that α, β, γ are wff’s. Show that the following statements are tautologies
(a) (α −→ (β −→ γ)) ←→ (α ∧ β −→ γ)
(b) (α ←→ (β ←→ γ)) ←→ (α −→ (β ←→ γ))
9. Express the following english sentences in first-order logic after giving suitable definitions
for the symbols used.
(a) Everybody that Spocks knows is illogical
(b) Nobody who knows Spock likes him
(c) Illogical people like everyone they know
(d) There is exactly one illogical person
10. Prove whether or not the following are valid wff’s. Assume P, Q, R are unary predicates
in some fixed language.
(a) (P (x) −→ ∃xQ(x)) ←→ ∃x(P (x) −→ Q(x))
(b) ∀x(P (x) −→ Q(y)) ←→ (∀xP (x) −→ Q(y))
(c) Assume x does not occur free in α: (α −→ ∀xβ) ←→ ∀x(α −→ β)
(d) (α −→ ∀xβ) ←→ ∀x(α −→ β)
11. Let M = < R, ·, 1 >, i.e. a structure with the universe the set of reals, multiplication
as a binary function and with the constant 1.
(a) Can you define {0}? If so, give a wff that defines it or show this can’t be done.
(b) Can you define {−1}? If so, give a wff that defines it or show this can’t be done.
(c) Can you define {2}? If so, give a wff that defines it or show this can’t be done.
(d) Can you define [0, ∞]? If so, give a wff that defines it or show this can’t be done.
(e) Can you define the function +? If so, give a wff that defines it or show this can’t
be done.
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