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

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

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

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<strong>MATH</strong> <strong>160</strong>, <strong>MATH</strong>EMATICAL <strong>LOGIC</strong> <strong>Midterm</strong> <strong>Review</strong> questions, Spring 2007.<br />

1. Either prove by using truth tables or disprove by giving a counter example (a particular<br />

truth assignment where the implication fails)<br />

(a) α −→ β |= β −→ α<br />

(b) (α ∨ β) ∧ γ |= α ∨ (β ∧ γ)<br />

2. Let φ be the wff (α −→ β) −→ γ<br />

(a) Find a wff tautologically equivalent to φ which uses only the connectives ¬ and ∧.<br />

(b) Find another wff tautologically equivalent to φ which is in disjunctive normal form.<br />

3. Let ⊕ denote exclusive or.<br />

(a) Is α ∧ ¬α (a contradiction) expressible with just {∨, ∧, ⊕}. If so, how ? If not,<br />

explain why ?<br />

(b) Is {∨, ∧, ⊕} complete? Prove your answer. Your argument needs to be convincing,<br />

but obvious facts do not need detailed proofs.<br />

4. Give a proof from the axioms of propositional calculus of the formula ¬(α −→ ¬β) −→<br />

α<br />

5. Let P C denote the theory of propositional calculus (= set of wff’s provable from axiom<br />

schemes L 1 , L 2 , L 3 using Modus Ponens) Let P C + be the extension of P C obtained by<br />

adding a fourth axiom scheme<br />

(¬α −→ β) −→ (α −→ ¬β)<br />

(a) Show that P C is a proper subset of P C + .<br />

(b) We say a theory is consistent if there is a wff α such that α is not provable. Is<br />

P C + consistent? Prove your answer. Hint: you might try γ −→ γ in place of α and β<br />

in the scheme.<br />

6. Let the formula σ i be defined as follows:<br />

Now define<br />

σ i = (P 1 ∨ P 2 ∨ . . . ∨ P i ) −→ P i+1 , i ≥ 1<br />

Σ = {σ i : i = 1, 2, 3, . . .}<br />

Describe all truth assignments ν that satisfy Σ.<br />

7. Let P C ′ be the theory in the language of propositional calculus which differs from P C<br />

only in having the axiom scheme<br />

L ′ 3 : (¬α −→ ¬β) −→ ((¬α −→ β) −→ α))<br />

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<br />

(a) (α −→ (β −→ γ)) ←→ (α ∧ β −→ γ)<br />

(b) (α ←→ (β ←→ γ)) ←→ (α −→ (β ←→ γ))<br />

9. Express the following english sentences in first-order logic after giving suitable definitions<br />

for the symbols used.<br />

(a) Everybody that Spocks knows is illogical<br />

(b) Nobody who knows Spock likes him<br />

(c) Illogical people like everyone they know<br />

(d) There is exactly one illogical person<br />

10. Prove whether or not the following are valid wff’s. Assume P, Q, R are unary predicates<br />

in some fixed language.<br />

(a) (P (x) −→ ∃xQ(x)) ←→ ∃x(P (x) −→ Q(x))<br />

(b) ∀x(P (x) −→ Q(y)) ←→ (∀xP (x) −→ Q(y))<br />

(c) Assume x does not occur free in α: (α −→ ∀xβ) ←→ ∀x(α −→ β)<br />

(d) (α −→ ∀xβ) ←→ ∀x(α −→ β)<br />

11. Let M = < R, ·, 1 >, i.e. a structure with the universe the set of reals, multiplication<br />

as a binary function and with the constant 1.<br />

(a) Can you define {0}? If so, give a wff that defines it or show this can’t be done.<br />

(b) Can you define {−1}? If so, give a wff that defines it or show this can’t be done.<br />

(c) Can you define {2}? If so, give a wff that defines it or show this can’t be done.<br />

(d) Can you define [0, ∞]? If so, give a wff that defines it or show this can’t be done.<br />

(e) Can you define the function +? If so, give a wff that defines it or show this can’t<br />

be done.<br />

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