Question

- Prove equivalent:

P⊃ (Q ⊃ P) and (~Q ⊃ (P ⊃ (~Q V P)))

Answer #1

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

Prove: ~p v q |- p -> q by natural deduction

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

Using rules of inference prove.
(P -> R) -> ( (Q -> R) -> ((P v Q) -> R) )
Justify each step using rules of inference.

are
they logically equivalent (show how) truth table or in word::
a) p —> ( q —> r ) and ( p -> q) —> r
b) p^ (q v r ) and ( p ^ q) v ( p ^ r )

[16pt] Which of the following formulas are semantically
equivalent to p → (q ∨ r): For each
formula from the following (denoted by
X) that is equivalent to p → (q
∨ r), prove the validity of X
« p → (q ∨
r) using natural deduction. For
each formula that is 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...

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)

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

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 the following: (By contradiction)
If p,q are rational numbers, with p<q, then there exists a
rational number x with p<x<q.

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