BadAtDiscrete
New member
- Joined
- Apr 16, 2022
- Messages
- 1
Find an explicit bijection f(x) from the set A = { 6, 12, 18, 24, …………} to the set
B = {0, 5, 10, 15, 20, ……}. Prove that your function is a bijection. Find the inverse of this function. For your function find f(2022) and find a x such that f(x) = 10000
2. Let A be a 5 by 10 matrix, B be a 10 by 100 matrix and C be a 100 by 1000 matrix. Find the number of operations (each addition is an operation and each multiplication is an operation) in finding AB, BC, A(BC) and (AB)C. You will not get any credit if you simply write a bunch of numbers. Explain how you did the calculations and arrived at your answer.
3. Without using a calculator find the prime factorization of 18! (this is 18 factorial)
4. Find the gcd(252, 198) and write it as a linear combination of 252 and 198 i.e. find the Bezout coefficients. You can use your calculator for numerical computations. You need to show all your work.
5. Assume the result: 1 if p is prime and if p does not divide a then 1mod( ) pa p −
Using this result and other properties of modular arithmetic find the values of
. You will not get any credit if you simply write an answer. 2022 1002 14 mod11 and 23 mod41
You need to explain how you used the above result to find your answer.
6. Using mathematical induction, for all , prove that 0 n 2 2 2 2 ( 1)(2 1)(2 3) 1 3 5 ...... (2 1) 3 n n n n + + + + + + + + =
7. Using mathematical induction, for all , prove that 1 n ( ) 1 2 1 21 is a divisor of 4 5 nn+ − +
8. Without using a calculator (except for simple arithmetic computations) find the binary, octal and hexadecimal representation of 1008. Show all the steps.
9. Find the gcd(100, 221) and write it as a linear combination of 100 and 221. Using this solve the congruence 1001mod(221) and 1007mod(221) x x
10. Prove that there is no onto function from { 1, 2, 3, 4, ……} to the interval ( 5, 6) = set of all real numbers between 5 and 6. =
B = {0, 5, 10, 15, 20, ……}. Prove that your function is a bijection. Find the inverse of this function. For your function find f(2022) and find a x such that f(x) = 10000
2. Let A be a 5 by 10 matrix, B be a 10 by 100 matrix and C be a 100 by 1000 matrix. Find the number of operations (each addition is an operation and each multiplication is an operation) in finding AB, BC, A(BC) and (AB)C. You will not get any credit if you simply write a bunch of numbers. Explain how you did the calculations and arrived at your answer.
3. Without using a calculator find the prime factorization of 18! (this is 18 factorial)
4. Find the gcd(252, 198) and write it as a linear combination of 252 and 198 i.e. find the Bezout coefficients. You can use your calculator for numerical computations. You need to show all your work.
5. Assume the result: 1 if p is prime and if p does not divide a then 1mod( ) pa p −
Using this result and other properties of modular arithmetic find the values of
. You will not get any credit if you simply write an answer. 2022 1002 14 mod11 and 23 mod41
You need to explain how you used the above result to find your answer.
6. Using mathematical induction, for all , prove that 0 n 2 2 2 2 ( 1)(2 1)(2 3) 1 3 5 ...... (2 1) 3 n n n n + + + + + + + + =
7. Using mathematical induction, for all , prove that 1 n ( ) 1 2 1 21 is a divisor of 4 5 nn+ − +
8. Without using a calculator (except for simple arithmetic computations) find the binary, octal and hexadecimal representation of 1008. Show all the steps.
9. Find the gcd(100, 221) and write it as a linear combination of 100 and 221. Using this solve the congruence 1001mod(221) and 1007mod(221) x x
10. Prove that there is no onto function from { 1, 2, 3, 4, ……} to the interval ( 5, 6) = set of all real numbers between 5 and 6. =
