4. [9 pts Let Q denote the rationals and IRQ the irrationals. Prove that (a) λ(Q)-0, (b) λ(in [0,1)-1.
1. For n-pg, where p and q are distinct odd primes, define (p-1)(q-1) λ(n) gcd(-1-1.411) Suppose that we modify the RSA cryptosystem by requiring that ed 1 mod X(n). a. Prove that encryption and decryption are still inverse operations in this modified cryptosystem. RSA cryptosystem.
P3. Define the hyperbolic distance between P and to be dH(P,Q) Ina, where a is defined in P2. Prove that da(P,) d(TP, TQ), where T - To,o is a horizontal translation. P3. Define the hyperbolic distance between P and to be dH(P,Q) Ina, where a is defined in P2. Prove that da(P,) d(TP, TQ), where T - To,o is a horizontal translation.
2. Consider the following relations defined from Λ to B, where Λ and B are defined as indicated. In each case, prove or disprove that f is a function from Λ to B. If f is a function frorn Л to B, determine whether or not the notation f:A-> B can be used. If not, how could Λ be changed to make the notation correct? 1+22- (g) Λ-R, l-R, and f(x) V1-3. 0 if Q 1 if 2 ERIQ (j)...
Express the statement (p → q) Λ (q Λ r) in disjunctive normal form. (¬р Λ q Λ r) (¬р Λ q Λ r) V (р Λ ¬q Λ r) V (р Λ q Λ r) (р Λ q Λ r) V (¬р Λ q Λ r) (¬р Λ q Λ r) V (р Λ ¬q Λ r)
+o0 P(A,) 0(n N4 0, 2. Let A A = Q , prove i1 +o0 P(A,) 0(n N4 0, 2. Let A A = Q , prove i1
Prove the validity of the following sequents in predicate logic, where F, G, P, and Q have arity 1, and S has arity 0 (a ‘propositional atom’):
5. The Exponential(A) distribution has density f(x) = for x<0' where λ > 0 (a) Show/of(x) dr-1. (b) Find F(x). Of course there is a separate answer for x 2 0 and x <0 (c Let X have an exponential density with parameter λ > 0 Prove the 'Inemoryless" property: P(X > t + s|X > s) = P(X > t) for t > 0 and s > 0. For example, the probability that the conversation lasts at least t...
Let z, y > 0. Prove that for any ε > 0 and p, q > 1 so that = 1 we have q + is 1s known as Young's inequality with E. Let z, y > 0. Prove that for any ε > 0 and p, q > 1 so that = 1 we have q + is 1s known as Young's inequality with E.
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q) is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP + βQ|α, β ∈ Z[x]}. (iii) For which primes p and which integers n ≥ 1 is the polynomial xn − p...