Discrete Mathematics Answers

4. Suppose p is the statement 'I play softball' and q is the statement 'The moon is 250,000 miles from Earth.' Select
the correct statement corresponding to the symbols ┐p ∧ q.
A. I don't play softball and the moon is 250,000 miles from Earth.
B. It is not the case that either I play softball or the moon is 250,000 miles from Earth.
C. I don't play softball or the moon is 250,000 miles from Earth.
D. It is not the case that both I play softball and the moon is 250,000 miles from Earth.

5. Select the statement that is the negation of “Coach Spurrier is charming and Coach Spurrier is modest.”
A. Coach Spurrier is not charming and Coach Spurrier is not modest.
B. Coach Spurrier is not charming or Coach Spurrier is not modest.
C. Coach Spurrier is not charming and Coach Spurrier is modest.
D. Let’s get serious for a minute.

Define the following terms with proper diagrams:

a. Degree of a node in a graph
b. Copies of binary trees

Define the following terms with proper example:

a. Relatively prime integers
b. Hadamard Matrix

Explain graph isomorphism with proper diagrams.

The recursive definition of a function X is given as:
f(0)=5 and f(n)=f(n-2)+5
Now, find out the value of f(14) using the above function.

Show that if A and B are sets with the same cardinality, then |A|<=|B| and |B|<=|A|

Prove that if n is an odd positive integer, then n^2 ≡ 1 (mod 8).

(19) Show that if n is an integer then n^2 ≡ 0 or 1 (mod 4).
(20) Use the result of Exercise (19) to show that if m is a positive integer that can be written in the form m= 4k+ 3 (where k is a nonnegative integer), then m is not the sum of the squares of two integers.

Show that if n|m, where n and m are integers greater than 1, and if a≡b(mod m), where a and b are integers, then a≡b(mod n).

Show that if a≡b(mod m) and c≡d(mod m), where a, b, c, d ϵ Z with m≥2, then a-c≡b-d(mod m).