Question

Prove or disprove that for any events A and B,

P(A) + P(B) − 1 ≤ P(A ∩ B) ≤ min{P(A), P(B)}.

Answer #1

Suppose that A, B and C are events. Prove or disprove the
statement “A, B and C are mutually exclusive if and only if A,
B and C are exhaustive”.

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

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 ).

1. Prove that {2k+1: k ∈ Z}={2k+3 : k ∈ Z}
2. Prove/disprove: if p and q are prime numbers and p < q,
then 2p + q^2 is odd (Hint: all prime numbers greater than 2 are
odd)

For Problems #5 – #9, you willl either be asked to prove a
statement or disprove a statement, or decide if a statement is true
or false, then prove or disprove the statement. Prove statements
using only the definitions. DO NOT use any set identities or any
prior results whatsoever. Disprove false statements by giving
counterexample and explaining precisely why your counterexample
disproves the claim.
*********************************************************************************************************
(5) (12pts) Consider the < relation defined on R as usual, where
x <...

For two events A and B show that P (A∩B) ≥ P (A)+P (B)−1. (Hint:
Apply de Morgan’s law and then the Bonferroni inequality). Derive
below Results 1 to 4 from Axioms 1 to 3 given in Section 2.1.2 in
the textbook.
Result 1: P (Ac) = 1 − P(A)
Result 2 : For any two events A and B, P (A∪B) = P (A)+P (B)−P
(A∩B)
Result 3: For any two events A and B, P(A) = P(A ∩...

8. Let a, b be integers. (a) Prove or disprove: a|b ⇒ a ≤ b. (b)
Find a condition on a and/or b such that a|b ⇒ a ≤ b. Prove your
assertion! (c) Prove that if a, b are not both zero, and c is a
common divisor of a, b, then c ≤ gcd(a, b).

Question2
(a) Prove that P(EFc) = P(E) − P(EF)
(b) Prove that P(EcFc) = 1 − P(E) – P(F) +
P(EF)
(c) Show that the probability that exactly one of the events E or F
occurs is equal to
P(E)+P(F )−2P(EF )

Please type if possible.
1. For two events A and B show that P(A∩B) ≥
P(A)+P(B)−1. (Hint: Apply de Morgan’s law and then the Bonferroni
inequality).
2. Derive below Results 1 to 4 from Axioms 1 to
3 given in Section 2.1.2 in the textbook.
Result 1: P(Ac ) = 1 − P(A)
Result 2 : For any two events A and B, P(A∪B) =
P(A)+P(B)−P(A∩B)
Result 3: For any two events A and B, P(A) = P(A ∩ B)...

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