Truth Tables, Implications, Contrapositives and Converses

(a) Use truth tables to prove that an implication is always equivalent to its contrapositive. Site an example where this is so. (b) Use truth tables to prove that an implication may not be equivalent to its converse. Site an example where this is so.

Logical Equivalents

Determine whether (p --> q ) / (p --> r) and p --> (q / r) are logically equivalent. Show all work.

Basic Proof

Prove or disprove the following: If the integer n is divisible by 3, then (nxn) is divisible by 3. Show work.

Integer Proofs by Contradiction

Assume that n is a positive integer. Use the proof by contradiction method to prove: (a) If 7n + 4 is an even integer, then n is an even integer. (b) Prove: n is an even integer iff 7n + 4 is an even integer. (Note this is an if and only if (iff) statement.

Set Operations : Subtraction, Cartesian Products, Direct Sums and Intersection

Let A = {1,3,5} and B = {3,5}. Compute: (a) A - B (b) A X B (c) A (plus sign with a circle around it) B (d) A ∩ (B - A)

Operands : AND and XOR

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

Matrix Addition and Multiplication and Applying the Distributive Law

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.

One to One and Inverse 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