Discrete Mathematics Answers

Given the following statements as premises:
If he takes coffee, he does not drink milk.
He eats crackers only if he drinks milk.
He does not take soup unless he eats crackers.
At noon today, he had coffee.
Therefore he took soup at noon today.
Verify the validity of the statements.

Find the coefficient of x^18 in the expansion of (1-x-x^2)^10

Write down the converse of each of the following statements: (2)
i) If n =1 (mod 4) for a natural number n, then n = x^2+y^2 for two integers
x and y.

1. Discuss two examples on binary trees both quantitatively and qualitatively

1. Write the multisets of prime factors for the given numbers.
I. 160
II. 120
III. 250
2. Write the multiplicities of each element of multisets in part 2(1-I, ii,iii) separately.
3. Find the cardinalities of each multiset in part 2-1.

1. Let A and B be two non-empty finite sets. If cardinalities of the sets A, B, and A∩B are respectively 72, 28 and 13, then find the cardinality of the set A∪B.

2. If n(A-B)=45, n(AUB )=110 and n(A∩B)=15, then find n(B).

3. If n(A)=33, n(B)=36 and n(C)=28, find n(A∪B∪C).

The binary operation on Z given by x
mathrmast (y=1+xy) not in the question later appered when i pasted)
is______ but not associative
a.Associative
b.Commutative
c.Distributive
d.Additive

Prove that n ! > 2^n for n a positive integer greater than or equal to 4.
Prove that LHS = RHS

a) Provide an example of a relation T on set A = {1, c, 3} that is irreflexive and satisfies trichotomy. (What is Trichotomy)

b) Let S= {(1, 2), (3, 4), (2, 3)} be a relation on set B = {1, 2, 3, 4}.
Is S functional? Motivate your answer

c) Let C = {1, 2, a, b} and let R = {(1, 1), (a, b), (b, 2)} and S = {(2, 1), (a, 1), (b, b), (b, 2), (2, a)} be two relations on C
i) Determine R o S (S;R)
ii) Which ordered pairs must be added to R to make it a reflexive relation?

a) Given the propositions, and . Construct a Truth Table and a justifying a sentence to show these two propositions are logically equivalent. (8 marks)

b) Construct a logic circuit for ¬ (((¬x ∧ ¬y) ∧ z) ∧ (x ˅ y)). (8 marks)