Discrete Mathematics Answers

draw a graph that has the following vertices and edges.
V = {x1, x2, x3, x4} and E = {{x1, x3, }, {x1, x4}, {x2, x4}}

draw the graphs that have the following vertices and edges
V = {a, b, c}, and E = {{a, b}, {b, c}}.

1. Expand in a two-element universe (the elements are named 'a' and 'b')

(a) ~(x) ((Fx v Gy) v Ka)

(b) (x) ~ (Kx v Ka)

(c) (Ex) (Cy v (Fx --> ~Ga))

2. For the following wffs, indicate which variables are free and which are bound (you can use 'F' for free and 'B' for bound.) Make sure you draw a vertical line underneath each variable with the letters 'F' or 'B' at the bottom of each vertical line.

You can write the wffs with the vertical lines on a piece of paper and submit a photo of it.



(a) (x) (y) (z) ((Fxy --> Gxy) v (Hxa --> Hzbu))

(b) (Ex) (Ey) Hxxy --> (z) (Gz v Hx)

(c) (z) (Ex) (y) (Axayzw v Bxycuvz)

The contrapositive of 'there exist y€z such that p(y) is true' is ' there exist x€z such that p(x) is true'.
Is the statement true or false? Justify your answer.

A school bus is transporting athletes home from an athletics meeting. Of these athletes 15% took part in hurdles,20% took part in the long jump event 35%,took part in relay races,30% ran the 100-m sprint and 25% took part in the high-jump event. Explain how is this possible.

Write the converse and the contrapositive of the following conditional:

Conditional: If it is sunny then it is daylight.

Converse:

Contrapositive:

A school bus is transporting athletes home from an athletics meeting.of these athletes 15% took part in hurdles,20% took part in the long jump event ,35% took part in the relay races,30% ran 100- m sprint and 25% took part in high jump explain how is possible

et U be the set of positive integers 1, 2, 3, ...
etc., A be the set of odd positive integers and
B be the set of even positive integers. Verify
De Morgan's laws.

4. Use mathematical induction to prove that 1^3 + 2^3 + ... + n^3 =
=(n(n+1)/2)^2
for all integers n ≥ 1

Which of the following statements are true? Give reasons for your answers.
i) Set {p,q,r} and {p,r, q, p} are equal.