Consider the relations E9 and E12 on the set of integers defined as follows • aE9b precisely when a ≡ b (mod 9). • aE12b precisely when a ≡ b (mod 12). Let R = E9 ◦ E12 (a) Suppose that 2Rx. Prove that x ≡ 2 (mod 3) (b) Suppose that x ≡ 2 (mod 3). Prove that 2Rx

Consider the relations E9 and E12 on the set of integers defined as follows • aE9b precisely when a ≡ b (mod 9). • aE12b precisely when a ≡ b (mod 12). Let R = E9 ◦ E12 (a) Suppose that 2Rx. Prove that x ≡ 2 (mod 3) (b) Suppose that x ≡ 2 (mod 3). Prove that 2Rx

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Let a, b be integers. What does it mean for a = b (mod 17)? Describe the elements of Z17. Compute…

A: a ≡ b (mod 17) means that a − b = 17k for some k ∈ ℤ. ∴ b − a = −17k = 17(−k); henceb ≡ a (mod…

Q: Let a, b, n be positive integers. Prove that if a = b (mod n), then GCD(a, n) = GCD(b,n).

Q: 1. Show that k1601 = k (mod 17) for any k.

Q: Consider the sentence (3xP(x) ^ 3xQ(x)) 3x(P(x) ^ Q(x)). Which of the following is true? A. The…

A: Let P(x) : x is such that x ≡ 1 (mod 3) Q(x) : x is such that x ≡ 2 (mod 6) Then,…

Q: Let x E Z. Prove that either x2 = 0 (mod 4) or x2 = 1 (mod 4).

A: By the help of even and odd integer's we prove the result. The answer is given as follows :

Q: 1.Let a, b, and c be integers, where a≠0. If a | b and a | c, then a | (b + c) True or…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 5. Find an integer x between 0 and 776 such that x = 13 (mod 21) x = 5 (mod 37)

A: Set N = 21 × 37 = 777. Then , m1 = N/21 = 777/21 = 37 and m2 = N/37 = 777/37 = 21 Now we need to…

Q: Let x be an integer. Prove that, if mod3=2, then 3|(x2-1).

A: I hove used the theorem on congruent

Q: For any integer a, prove that a=0 or 1 (mod 3).

A: The solution is given as follows

Q: Solve the congruence 11x = 45 (mod 97) where x is an integer in the interval [0,97).

Q: Prove that the number of solutions to the congruence xk ≡ 1 mod p is gcd(k, p − 1)

Q: Let a, b, c and meZ, and m 2 2 1. Prove: if a =b (mod m), then GCF (a, m) = GCF (b, m) %3D m 2.…

Q: Let m e N and a E Z. Suppose that 3|(m) and that a is a primitive root (mod m). Prove that a is not…

Q: Suppose x F and you discover that x¹ = 1 mod p and x² = 1 mod p. Please prove that xgcd(m,n) = 1 mod…

Q: Let a, n be positive integers. Knowing that 15a – 13n = 1, we deduce that the multiplicative inverse…

A: We have to find the moduler multiplicative inverse of a.

Q: 1.21. Let m € Z. (a) Suppose that m is odd. What integer between 1 and m - 1 equals 2-¹ mod m? (b)…

Q: Exercise 1. Suppose that a and b are integers, a = 3(mod11) and b = 8 (mod11) and c = a3 - b3. Then…

Q: x

Q: Let a, b, c, m be integers with m > 2. Prove that if ac ≡ bc mod m and d = gcd(m, c) , then a ≡ b…

Q: let a,b,c,d, and n be integers with n>1. Prove that if a is congruent to b(mod n) and c is congruent…

Q: 2. Prove that if ar b (mod n), then () x = () (mod ()) where d = gcd(a, n). Apply this result to 5x…

A: We first find the gcd of the given numbers and then use that to solve the given congruence. The…

Q: Suppose that a ≡ 9(mod 13) and b ≡ 7(mod 13). Find the integer cwith0≤c≤12suchthatc≡2a2 −6b2(mod13).

Q: Let a and b be positive integers with (a, b) = 1. Show that if g is a primitive root (mod ab), then…

A: We have given (a,b)=1 where a, b both are positive integer. We have to show if g is primitive root…

Q: For which positive integers a is the congru

A: Given, ax4≡2 (mod 13)

Q: Show that if a is a quadratic residue (mod p) and ab = 1 (mod p) then b is a quadratic residue (mod…

A: Note that quadratic residues means those which are squares mod p using this definition we solve the…

Q: For each pair of integers m and d, calculate m mod d. Show your work! a. m = 43 and d = 9 b. m 50…

A: If m and d be two integers . Then m mod d means that the remainder when m is divided by d .

Q: 7. Let n E Z. Prove that 9n3 6 (mod 27).

A: The objective is to prove the above statement.

Q: Let c, d e Z and m e N. For each statement decide whether it is necessarily true, or whether it can…

Q: Prove that xk = x* mod |G| for x € G for any integer k.

A: Prove it considering two cases.

Q: 3. For the following statements, either prove that they are true or provide a counterexample: (a)…

A: Given that, Let a, b, m, n ∈ Z such that m, n > 1 and n | m. a ≡ b (mod m) By using the…

Q: Suppose that each of a, b, and c is an integer and m is a positive integer. Show that if a = b(mod…

Q: Determine all solutions x ∈ Z to the triple x ≡ 5 (mod 6), x ≡ 7 (mod 10) and x ≡ 2 (mod 15) of…

A: We reduce the congruence and then solve

Q: Let c, d e Z and m E N. For each statement decide whether it is necessarily true, or whether it can…

Q: true or false 1..The GCD of -50, -100 and -125 is -25 2.If (a, b) = d and d│m, then (ax is…

A: Solution

Question

Consider the relations E9 and E12 on the set of integers defined as follows

• aE9b precisely when a ≡ b (mod 9).
• aE12b precisely when a ≡ b (mod 12).

Let R = E9 ◦ E12

(a) Suppose that 2Rx. Prove that x ≡ 2 (mod 3)
(b) Suppose that x ≡ 2 (mod 3). Prove that 2Rx

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Trending now

This is a popular solution!

Step by step

Solved in 6 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

Show that if a is a quadratic residue (mod p) and ab = 1 (mod p) then b is a quadratic residue (mod p).
Find all integers X satisfying 13X ≡ 4 mod 2024. Show your working
Use the definition of congruence to prove that if j, k, 1, m, n E Z are such that k =1 (mod n) and j=m (mod n) then kj = Im (mod n).
Find all integers X satisfyingX ≡ 1 mod 6, X ≡ 3 mod 8, X ≡ 7 mod 9
1. Let J₁ = {0, 1,2,3} and define f: J₁ J₁ and g: J₁ J₁ by f(x) = (x + 3)² mod 4 and g(x)=x²-3x +- 5 mod 4. Is f = g? Explain your answer.
Let x be an integer. Prove that if x mod 3 is not 1, then 3|(x2+x).
Let Abe the set of integers and let x, y be integers. Furthermore, let Ry hold if and only if a = y(mod 5). The modulo operator, mod n (in this case, n = = 5), outputs the remainder when y is divided by n. Such a relation R is called congruence modulo n, or the '%' symbol in many computer programs. Answer the following: Is a = y( mod 5) reflexive? Select] Select] Yes Is a = y( mod 5) symmetric No Is a = y(mod 5) anti-symmetric? [Select] Isa = y( mod 5) transitive? Select]
Let f: N11 → N11 be defined by f(x) = (7x + 2) mod 11. Find f-¹ if it exists.
2.1.3. If a, b are integers such that a = b (mod p) for every positive prime p, prove that a = b.
Suppose gcd(a,n) = 1. Then aφ(n) ≡ 1 (mod n).
Nitin