Decide whether or not the following equivalences are valid. (a) [∀x,(P(x) ∨ Q(x))] ≡ [(∀x, P(x))...
Decide whether or not the following equivalences are valid.
(a) [∀x,(P(x) ∨ Q(x))] ≡ [(∀x, P(x)) ∨ (∀x, Q(x))].
(b) [∃x,(P(x) ∨ Q(x))] ≡ [(∃x, P(x)) ∨ (∃x, Q(x))].
For each, demonstrate the equivalence or give a counterexample.
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Decide whether or not the following equivalences are valid.
(a) [∀x,(P(x) ∨ Q(x))] ≡ [(∀x, P(x))...
(a) use the logical equivalences p → q ≡∼p ∨ q and p ↔ q ≡...
(a) use the logical equivalences p → q ≡∼p ∨ q and p ↔ q ≡ (∼p ∨ q) ∧ (∼q ∨ p) to rewrite the given statement forms without using the symbol → or ↔, and (b) use the logi- cal equivalence p ∨ q ≡∼(∼p∧ ∼q) to rewrite each statement form using only ∧ and ∼. * p∨∼q→r∨q
1. Use the DPP to decide whether the following sets of clauses are satisfiable. (a) {{¬Q,T},{P,¬Q},{¬Q,¬S},{¬P,¬R},{P,¬R...
1. Use the DPP to decide whether the following sets of clauses are satisfiable. (a) {{¬Q,T},{P,¬Q},{¬Q,¬S},{¬P,¬R},{P,¬R,S},{Q,S,¬T},{¬P,S,¬T},{Q,¬S},{Q,R,T}} (b) {{¬Q,R,T},{¬P,¬R},{¬P,S,¬T},{P,¬Q},{P,¬R,S},{Q,S,¬T},{¬Q,¬S},{¬Q,T}} 2. Decide whether each of the following arguments are valid by first converting to a question of satisfiability of clauses (see the Proposition), and then using the DPP. (Note that using DPP is not the easiest way to decide validity for these arguments, so you may want to use other methods to check your answers) (a) (P → Q), (Q → R),...
Question 6 (2 points). Decide whether the following argument is valid, using a truth tree: H (D(BV P), DVP Question 6 (2 points). Decide whether the following argument is valid, using a truth...
Question 6 (2 points). Decide whether the following argument is valid, using a truth tree: H (D(BV P), DVP
Question 6 (2 points). Decide whether the following argument is valid, using a truth tree: H (D(BV P), DVP
1. Use a truth table to find if the following is valid or not valid: p...
1. Use a truth table to find if the following is valid or not valid: p → r q → r q ˅ ¬r Therefore, ¬p Valid Not Valid Discrete Math 2. Indicate whether each expression is an equivalence of the following: p ˄ q p ˅ q p → q ¬(p → q) (p ˄ q) ˅ (p ˄ q) ¬ (¬p ˅ ¬q) 3. For the given values for p, q, and r,...
Verify the logical equivalences using the theorem below: (p ∧ ( ~ ( ~ p ∨...
Verify the logical equivalences using the theorem below:
(p ∧ ( ~ ( ~ p ∨ q ) ) ) ∨ (p ∧ q) ≡ p
Theorem 2.1.1 Let p, q, and r be statement variables, t a tautology, and c a contradiction. The following logical equivalences are true. 1. Commutativity: p1q=q1p; p V q = 9VP 2. Associativity: ( pq) Ar=p1qAr); (pVq) Vr=pv (Vr) 3. Distributivity: PA(Vr) = (p19) (par); p V (qar) = (pVg) (Vr) 4. Identity: pAt=p:...
Problem 3.11 Show using a chain of logical equivalences that (p → r)A(q → r) pv q) → Show transcribed image text
Problem 3.11 Show using a chain of logical equivalences that (p → r)A(q → r) pv q) →
Note that x does not occur free in C in this equivalence. Prove the following using other equivalences 4. ∃x(c → A(x)) ≡...
Note that x does not occur free in C in this equivalence. Prove the following using other equivalences 4. ∃x(c → A(x)) ≡ c → ∃xA(x) 5. ∀x(A(x) → C) ≡ ∃A(x) → C 6. ∃x(A(x) → C) ≡ ∀xA(x) → C
1. For each of the following, state whether it is true or false. If false, give...
1. For each of the following, state whether it is true or false. If false, give a counterexample. If true, give a convincing argument why it is true. For all a, B e Pimla, (a) if (a + B) is valid and a is valid, then ß is valid. (b) if (a + b) is satisfiable and a is satisfiable, then ß is satisfiable. (c) if (a + b) is valid and a is satisfiable, then ß is satisfiable.
(a) Determine whether the following argument is valid: p =r 9 + (pva) (b) Determine whether...
(a) Determine whether the following argument is valid: p =r 9 + (pva) (b) Determine whether the following argument is valid: pr 9 → (avr) .
3. (10 pts.) Use logical equivalences to show that (p r)v(q r) and (pAq) r ane...
3. (10 pts.) Use logical equivalences to show that (p r)v(q r) and (pAq) r ane logically equivalent.
Question 6 (2 points). Decide whether the following argument is valid, using a truth tree: H (D(BV P), DVP
Question 6 (2 points). Decide whether the following argument is valid, using a truth tree: H (D(BV P), DVP
Verify the logical equivalences using the theorem below:
(p ∧ ( ~ ( ~ p ∨ q ) ) ) ∨ (p ∧ q) ≡ p
Theorem 2.1.1 Let p, q, and r be statement variables, t a tautology, and c a contradiction. The following logical equivalences are true. 1. Commutativity: p1q=q1p; p V q = 9VP 2. Associativity: ( pq) Ar=p1qAr); (pVq) Vr=pv (Vr) 3. Distributivity: PA(Vr) = (p19) (par); p V (qar) = (pVg) (Vr) 4. Identity: pAt=p:...
Problem 3.11 Show using a chain of logical equivalences that (p → r)A(q → r) pv q) →
1. For each of the following, state whether it is true or false. If false, give a counterexample. If true, give a convincing argument why it is true. For all a, B e Pimla, (a) if (a + B) is valid and a is valid, then ß is valid. (b) if (a + b) is satisfiable and a is satisfiable, then ß is satisfiable. (c) if (a + b) is valid and a is satisfiable, then ß is satisfiable.
(a) Determine whether the following argument is valid: p =r 9 + (pva) (b) Determine whether the following argument is valid: pr 9 → (avr) .
3. (10 pts.) Use logical equivalences to show that (p r)v(q r) and (pAq) r ane logically equivalent.
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