Discrete Mathematics Answers

Let A = {a, 1, c, 3} and B = {-1, 0, 1}

(a) find the Cardinality of A

(b) P(A)

(c) A x B

Determine whether the argument is valid or invalidusing critical row.

a. 𝑝⋁~𝑞

𝑞 over ∴𝑝

b. 𝑝→~𝑞

𝑞 over ∴~𝑝

Instruction: Write each statement in symbolic form and determine whether the argument is valid or invalid using critical row

.

a. If Caroline is late (l), we will not pursue our plan (~p). We did not pursue our plan. Therefore, Caroline was late.

b. If you are Mathematician (m), then you are logical (l). You are mathematician. Therefore, you are logical.

c. He will attend PLM (p) or UP (u). He did not attend UP. Therefore, he attended PLM.

Explain how to use a zero-one matrix to represent a relation on a finite set

Question 2: Give an example of each of the following graph containing:

i.An Euler circuit but not Hamiltonian cycle

ii.A Hamiltonian cycle but not an Euleriancircuit

iii.An Eulerian circuit as well as Hamiltonian cycle

Question 2: Define Hamiltonian circuit and Euler circuit. Give an example of each of the following graphs

i.N-Cube (Q₃)

ii.Complete Bipartite

iii.Regular graph of degree 4

iv.Cycles graph (C4, C₅)

v.Wheels graph (W₄ ,W₅)

vi.Hamiltonian, but non- Eulerian

vii.Eulerian but non-Hamiltonian

viii.Eulerian as well as Hamiltonian.

Question 2: Let R be a relation in a set A, and derive from R another relation S in A as follows:

x S y if (x R y xor y R x).

Recall that xor, exclusive or, is defined as: p xor q is true if (p is true and q is false, or p is false and q is true).

a) Prove that S is irreflexive.

b) Prove that S is symmetric.

Question 1: Let R be a relation in a set A, and derive from R another relation S in A as follows:

x S y if (x R y or y R x).

a) Prove that if R is reflexive, then S is reflexive.

b) Prove that S is symmetric.

c) Prove that if R is transitive, S is not necessarily transitive (by a counterexample).

d) If R is antisymmetric, is S antisymmetric? Prove your answer.

e) If R is an equivalence relation, is S an equivalence relation? Prove your answer.

f) If R is a partial order, is S a partial order? Prove your answer.