Question

- [16pt] Which of the following formulas are semantically
equivalent to
*p → (q*∨*r):*Forfrom the following (denoted by*each formula*) that is equivalent to*X**p → (q*∨*r),*prove the validity of*X*∨*« p → (q**r**)*using natural deduction. Forthat is*each formula***not**equivalent to*p → (q*∨*r),*draw its truth table and**clearly mark**the entries that result in the inequivalence.*Assume the binding priority used in class.*

(2.1) *q* ∨ *(¬p* ∨
*r**)*

(2.2) *q* ∧ *¬r → p*

(2.3) *p* ∧ *¬r → q*

(2.4) *¬q* ∧ *¬r →
¬p*

Answer #1

(2.1)Given:

In row 4 and row6 both are not equal.

Prove: (p ∧ ¬r → q) and p → (q ∨ r) are biconditional using
natural deduction NOT TRUTH TABLE

1. Construct a truth table for: (¬p ∨ (p → ¬q)) → (¬p ∨ ¬q)
2. Give a proof using logical equivalences that
(p → q) ∨ (q →
r) and (p → r)
are not logically equivalent.
3.Show using a truth table that (p →
q) and (¬q →
¬p) are logically equivalent.
4. Use the rules of inference to prove that the premise
p ∧ (p
→ ¬q) implies
the conclusion ¬q. Number each
step and give the...

Establish the equivalence of the following formulas:
a) ┐(p v q), ┐p ^ ┐q
b) p v (q ^ r), (p v q) ^ (p v r)
c) p -> (q -> r), (p ^ q) -> r
d) p <-> q, (p -> q) ^ (q -> p)

I. WHICH OF THE FOLLOWING FORMULAS ARE NOT WFFS ? _____1. ~q ~p
_____2. É(q · r) _____3. p v q · ~r É q _____4. (p v r) · (p v q) É
(r · ~q) _____5. (r v s) v ~[(p É r) · (s v q)] _____6. (p · q) v
(~r É p) _____7. ~(p É r · p É q) _____8. s _____9. ~~~r _____10.
~(p v r) · q

Convert (and simplify) the following sentences to Conjunctive
Normal Form (CNF):
2.1. (P →Q) → ((Q → R) → (P → R))
2.2. (P → Q) ↔ (P → R)

(1) Determine whether the propositions p → (q ∨ ¬r) and (p ∧ ¬q)
→ ¬r are logically equivalent using either a truth table or laws of
logic.
(2) Let A, B and C be sets. If a is the proposition “x ∈ A”, b
is the proposition “x ∈ B” and
c is the proposition “x ∈ C”, write down a proposition involving a,
b and c that is logically equivalentto“x∈A∪(B−C)”.
(3) Consider the statement ∀x∃y¬P(x,y). Write down a...

For three statements P, Q and R, use truth tables to verify the
following.
(a) (P ⇒ Q) ∧ (P ⇒ R) ≡ P ⇒ (Q ∧ R).
(c) (P ⇒ Q) ∨ (P ⇒ R) ≡ P ⇒ (Q ∨ R).
(e) (P ⇒ Q) ∧ (Q ⇒ R) ≡ P ⇒ R.

The mean of a binomial distribution is found using which of the
following formulas?
p*q
q*x
n*p
p*x

The implication - (p --> -q) is equivalent to which of the
following?
- p V -q
- p V q
p ^ -q
- q --> p

Use a truth table to determine whether the following argument is
valid.
p
→q ∨ ∼r
q →
p ∧ r
∴ p →r

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