Question

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

Answer #1

Are the statement forms P∨((Q∧R)∨ S) and ¬((¬ P)∧(¬(Q∧ R)∧ (¬
S))) logically equivalent? I found that they were not logically
equivalent but wanted to check. Also, does the negation outside the
parenthesis on the second statement form cancel out with the
negation in front of P and in front of (Q∧ R)∧ (¬ S)) ?

1) Show that ¬p → (q → r) and q → (p ∨ r) are logically
equivalent. No truth table and please state what law you're using.
Also, please write neat and clear. Thanks
2) .Show that (p ∨ q) ∧ (¬p ∨ r) → (q ∨ r) is a tautology. No
truth table and please state what law you're using. Also, please
write neat and clear.

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

Prove or disprove: If the columns of B(n×p) ? R are linearly
independent as well as those of A, then so are the columns of AB
(for A(m×n) ? R ).

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

Prove or disprove: The group Q∗ is cyclic.

Prove equivalent:
P⊃ (Q ⊃ P) and (~Q ⊃ (P ⊃ (~Q V
P)))

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

1. Prove p∧q=q∧p
2. Prove[((∀x)P(x))∧((∀x)Q(x))]→[(∀x)(P(x)∧Q(x))]. Remember to
be strict in your treatment of quantifiers
.3. Prove R∪(S∩T) = (R∪S)∩(R∪T).
4.Consider the relation R={(x,y)∈R×R||x−y|≤1} on Z. Show that
this relation is reflexive and symmetric but not transitive.

Prove or disprove that for any events A and B,
P(A) + P(B) − 1 ≤ P(A ∩ B) ≤ min{P(A), P(B)}.

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