Question

Prove that Pr[A] ≤ min(1, q/p) when Pr[B|A] ≥ p > 0 and Pr[B] ≤ q

Answer #1

in this problem we have to prove the following probability as follows :-

Find the volume of the parallelepiped with adjacent edges
PQ, PR, PS.
P(3, 0, 3), Q(−1, 2,
8), R(6, 1, 1), S(2,
6, 6)

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

The entity X ~ N(0, 1) (Standard normal distribution). (a) Find
q such that Pr(X <= q) = 0.25 (the first quartile of the
standard normal distribution); (b) Find q such that Pr(X <= q) =
0.75 (third quartile of the standard normal distribution)

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

Find the volume of the parallelepiped with adjacent edges
PQ, PR,
and PS.
P(−2, 1, 0), Q(3, 3, 3), R(1, 4, −1), S(3, 6, 2)

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

Let p and q be primes. Prove that pq + 1 is a square if and only
if p and q are twin primes. (Recall p and q are twin primes if p
and q are primes and q = p + 2.) (abstract algebra)

(§2.1) Let a,b,p,n ∈Z with n > 1.
(a) Prove or disprove: If ab ≡ 0 (mod n), then a ≡ 0 (mod n) or
b ≡ 0 (mod n).
(b) Prove or disprove: Suppose p is a positive prime. If ab ≡ 0
(mod p), then a ≡ 0 (mod p) or b ≡ 0 (mod p).

Prove p ∨ (q ∧ r) ⇒ (p ∨ q) ∧ (p ∨ r) by constructing a proof
tree whose premise is p∨(q∧r) and whose conclusion is
(p∨q)∧(p∨r).

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