Question

Given: (P & ~ R) > (~R & Q), Q> ~P Derive: P > R. use propositional logic and natural derivation rules.

Answer #1

Use the laws of propositional logic to prove the following:
1) (p ∧ q ∧ ¬r) ∨ (p ∧ ¬q ∧ ¬r) ≡ p ∧ ¬r
2) (p ∧ q) → r ≡ (p ∧ ¬r) → ¬q

p → q, r → s ⊢ p ∨ r → q ∨ s
Solve using natural deduction rules.

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

Given statement p q, the statement q p is called its converse.
Let A be the statement: If it is Tuesday, then you come to campus.
(1) Write down the converse of A in English. (2) Is the converse of
A true? Explain. (3) Write down the contrapositive of A in
English
Let the following statement be given: p = “You cannot swim” q =
“You are less than 10 years old” r = “You are with your parents”
(1)...

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.

4 Inference proofs
Use laws of equivalence and inference rules to show how you can
derive the conclusions from the given premises. Be sure to cite the
rule used at each line and the line numbers of the hypotheses used
for each rule.
a) Givens:
1 a∧b
2 c → ¬a
3 c∨d
Conclusion: d
b) Givens
1 p→(q∧r)
2 ¬r
Conclusion ¬p

What is the correct meaning of the logical expression p→q∨r∧s
?
((p→q)∨r)∧s
p→((q∨r)∧s)
(p→(q∨r))∧s
p→(q∨(r∧s))

Prove
a)p→q, r→s⊢p∨r→q∨s
b)(p ∨ (q → p)) ∧ q ⊢ p

Prove or disprove that [(p → q) ∧ (p → r)] and [p→ (q ∧ r)] are
logically equivalent.

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

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