
Prove pc=p (the charpoly of C coincides with the polynomial
p)
0 ao 01 C: prove that pc p. Hint: To prove that the charpoly of C coincides with the polynomial p.
0 ao 01 C: prove that pc p. Hint: To prove that the charpoly of C coincides with the polynomial p.
2.72 Prove that P(A'nB) 1P(An B) P(A)- P(B)
g(p+1)/2 (a) Suppose 9 is a p rimitive root of an odd prime p. Prove that- (mod p)
g(p+1)/2 (a) Suppose 9 is a p rimitive root of an odd prime p. Prove that- (mod p)
t e omaim ws R Prove meules Prove cule ibeth P, and pa P ave Presedive Projedve R- mo
t e omaim ws R Prove meules Prove cule ibeth P, and pa P ave Presedive Projedve R- mo
Prove that P(A' n B') = 1 + P(A n B)- P(A)- P(B)
Prove that that if p is a prime such P = 1 (mod 4), then (972) != -1 (mod P).
a) Prove algebraically that(m+n | p+n)≥(m | p) for all m, p, n ∈
N and such that m≥p.
b) Prove the above inequality by providing a combinatorial
proof. Hint: this can be done by creating a story to count the RHS
exactly (and explain why that count is correct), and then providing
justification as to why the LHS counts a larger number of
options.
a) Prove algebraically that p for all m, p, n EN, and such that m...
10. Prove that P(E UFUG)P(E) P(F) + 2P(EFG). 11. If P(E)9 and P(F).8, show that P(EF .7 In general, prove Bonferroni's inequality, namely, P(EF) 2 P(E) + P(F)-1 13. Prove that P(EF*)= P(E)-P(EF).
1. Prove“inclusion-exclusion,”thatP(A∪B)=P(A)+P(B)−P(A∩B). 2. Prove the “unionbound, ”thatP(A1∪A2)≤P(A1)+P(A2). Under what conditions does the equality hold? 3. Provethat, for A1 andA2 disjoint, P(A1∪A2|B)=P(A1|B)+P(A2|B). 4. A and B are independent events with nonzero probability. Prove whether or not A and Bc are independent.
Assume that p NAND q is logically equivalent to ¬(p ∧ q). Then, (a) prove that {NAND} is functionally complete, i.e., any propositional formula is equivalent to one whose only connective is NAND. Now, (b) prove that any propositional formula is equivalent to one whose only connectives are XOR and AND, along with the constant TRUE. Prove these using a series of logical equivalences.