Proof and Converse

Let A, B, and C be any three sets and consider the sentence: If A ∩ B = A ∩ C , then B = C. (a) Prove in detail that this statement is false. (Give a counterexample) (b)Write the converse of the above statement and show through an example that it is true (no proof necessary).

Simple Math

What is the value of x after each of the following statements are encountered in a computer program, if x = 1 before the statement is reached. Explain fully. (a) if 2 + 3 = 6 AND 3 + 4 = 7 then x:= x + 1 (b) if 2 + 3 = 6 XOR 3 + 4 = 7 then x:= x + 1

Rule of Products

A bit string is a string of bits (0’s and 1’s). The length of a bit string is the number of bits in the string. An example, of a bit string of length four is 0010. An example, of a bit string of length five is 11010. Use the Rule of Products to determine the following: (a) How many bit strings are there of length ...continues

Distributed Law for Matrices

Compute: (a) AC + BC (It is much faster if you use the distributive law for matrices first.) (b) 2A - 3A See attached file for full problem description.

Basic Matrix Laws

Let A and B be arbitrary n x n matrices whose entries are real numbers. Use basic matrix laws only to expand (A + B)². Explain all steps. Hint: Use the distributive laws.

Functions

Let A = {1,2,3} and B = {a,b,c}, and let f: A B. (a) Give an example of a one to one function from A to B (use the given sets A and B above). Briefly explain why your example is a 1-1 (one-to-one) function. (b) How many one to one functions from A to B are there? Explain. (c) Using the above sets A and B define ...continues

System of equations whose matrix is row equivalent

System of equations whose matrix is row equivalent. See attached file for full problem description.

Solve the following systems of equations

Solve the following systems of equations. See attached file for full problem description.

Show that this matrix is the inverse of...

Show that: (Matrix) 2 3 -1 1 2 1 -1 –1 3 is the inverse of 7 -8 5 -4 5 3 1 -1 1 Show all work.

Statement, Proof, Inductive Hypothesis

Let p(n) be the statement that: 1^3 + 2^3 + ... + n^3 = (n (n + 1) /2)^2 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of the proof. c) What is the inductive hypothesis? d) What do you need to prove in the inductive step? e) Complete the inductive ...continues