Give an example, in English, where it is true that an implication is always equivalent to its contrapositive. Also, an example in English of where an implication may not be equivalent to its converse

Give an example, in English, where it is true that an implication is always equivalent to its contrapositive. Also, an example in English of where an implication may not be equivalent to its converse.

Discrete Structures

In basic algebra the following Theorem is used frequently. If x,y and z are any three real numbers and if x + z = y + z then x = y. The analogous statement for sets would read: Let A, B, and C be any three sets. If A B = A C then B = C. Prove in detail that this statement is false. (Hint: Give a counterex ...continues

Proof by Contradiction

Assume that n is a positive integer. Use the proof by contradiction method to prove: If 7n + 4 is an even integer then n is an even integer. State your assumption(s) and what you wish to prove clearly.

Discrete Math

In basic algebra the following Theorem is used frequently. If x,y and z are any three real numbers and if x + z = y + z then x = y. The analogous statement for sets would read: Let A, B, and C be any three sets. If A union B = A union C then B = C. Prove in detail that this statement is false. (Hint: Giv ...continues

English to Logic Translation

Express the following sentence in logic and the write its negative. Be careful, use the rules of logic. "All people taking this class are under 30 years old and wear glasses."

Fuzzy Logic

Truth Values that are between 0 and 1 indicate varying degrees of truth. For instance, the truth value 0.8 can be assigned to the statement "Bob is happy," because Bob is happy most of the time, and the truth value 0.4 can be assigned to the statement "Max is happy, "because Max is happy slightly less than half the time. Th ...continues

Rules of Products

How many rows are there for any logical expression? containing: (i) Two (logical) variables? Why is the solution 2^2 ? What would be an example of this? Determine the number of different (that is nonequivalent) Boolean functions (logical expressions) containing: (ii) Two Variables? Why is the Solution 2^2^² ? What would be ...continues

Discrete Functions : Inverses and One-to-one Functions

Please see the attached file for the fully formatted problems. (a) Define the function f: R R by f(x) = x2 - 6. Briefly explain why f is not a 1-1 (one-to-one) function. (b) Is the function g: Z Z defined by g(n) = a one to one function? (Be careful, means the ceiling function.) Explain. (c) Briefly exp ...continues

Matrix Operations

Please see the attached file for the fully formatted problems. Let A = , B = and C = Compute: (a) AC + BC (b) 2A - 3A (c) Perform the given operation for the following zero-one matrices. .

Gauss-Jordan Technique

Solve the following systems of equations using the Gauss-Jordan technique x1 + x2 = 1 -x1 + x2 + x3 = -1 -1x2 + x3 = 3