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.
Homework Answers
Add Answer to:
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
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.
(40 pts) Consider the transitivity of the biconditional: ((P HQ ) ^ ( Q R ))...
(40 pts) Consider the transitivity of the biconditional: ((P HQ ) ^ ( Q R )) → ( P R ). a. Show that this argument is valid by deriving a tautology from it, a la section 2.1. You may use logical equivalences (p. 35), the definition of biconditional, and the transitivity of conditional: (( P Q) ^ (Q→ R)) → (P + R). Show that this argument is valid using a truth table. Please circle the critical lines. C....
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.
(40 pts) Consider the transitivity of the biconditional: ((P HQ ) ^ ( Q R )) → ( P R ). a. Show that this argument is valid by deriving a tautology from it, a la section 2.1. You may use logical equivalences (p. 35), the definition of biconditional, and the transitivity of conditional: (( P Q) ^ (Q→ R)) → (P + R). Show that this argument is valid using a truth table. Please circle the critical lines. C....
Most questions answered within 3 hours.
-
Calculate whether loci A&B are in linkage disequilibrium
(D) using the following data. Is the haplotype...
asked 6 minutes ago -
Barrington Box Enterprises has two divisions, large and small,
that share the common costs of the...
asked 9 minutes ago -
An integer number is said to be a perfect number if it is equal
to the...
asked 21 minutes ago -
What does economies of scale often fail to measure? How would
you recommend addressing this limitation?
asked 20 minutes ago -
Joanne Quick made an investment of $18,787.14. From this
investment, she will receive $2,000 annually for...
asked 28 minutes ago -
Enter your answer in the provided box. The solubility of CO2 in
water at 25°C and...
asked 34 minutes ago -
Amazon as with other ecommerce retailers struggle with cross
border solutions and continue to be country...
asked 33 minutes ago -
A second baseman tosses the ball to the first baseman, who
catches it at the same...
asked 50 minutes ago -
An object rests flat on a tablecloth, which is on top of a
table. Suppose you...
asked 50 minutes ago -
Which of the following will most likely occur as the result of
an unanticipated increase in...
asked 46 minutes ago -
Adria Lopez, owner of Success Systems, decides to prepare a
statement of cash flows for her...
asked 1 hour ago -
The reach and involvement of the federal government, in terms of
its role in disaster relief,...
asked 1 hour ago