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Show that (~𝒑∨𝒒)∧(𝒑∧~𝒒) is a contradiction.

Construct a graph G with 6 vertices {v1, v2, v3, v4, v5, v6} and six edges {e1, e2, e3, e4, e5,

e6} such that

i. e2 is a loop at v2

ii. v2 and v5 are end point of e5

iii. v3 is adjacent to v2

iv. v4 is isolated

v. e3 is parallel to e5

vi. e4 is incident of v1 and v6

Determine whether each of these functions from Z to Z is one-to-one.

a. f(n) = n2

+ 1

b. f(n) = n

Negation of statement (A ∧ B) → (B ∧ C) is:  (A ∧ B) →(~B ∧ ~C)

Can you explain the solution please?

Let n be your special number. Let p be the smallest prime divisor of n. Consider the complete bipartite graph Kp,n. (a) Does Kp,n have a Hamilton circuit? If so, describe it. If not, explain why not. (b) Does Kp,n have a non-cyclic Hamilton path? If so, describe it. If not, explain why not. (c) Does Kp,n have an Euler cycle? Explain your answer. (d) Does Kp,n have a non-cyclic Euler path? Explain your answer.

15. I provide 5 chairs, one for each person that is going to wait in the line. In how many different ways can they stand in line?

2. Prove that a relation R on a set A is symmetric if R-1 = R.

3. Give an example of a relation that is reflexive but neither symmetric nor transitive.

4. Show that the relation ‘is perpendicular to’ over the set of all straight lines in the plane is symmetric but neither reflexive nor transitive.

5. Let S and T be sets with m and n element respectively. How many elements has S × T? How many relations are there in S × T?

6. If R and S are equivalence relations in the set X, prove that R S is an equivalence relation.

7. Show that the relation of congruence modulo m has m distinct equivalence classes.

8. Show that a partition of a set S deter- mines an equivalence relation in S.

9. Let S = {n: n N and n > 1}. If a, b S define a ~ b to mean that a and b have the same number of positive prime factors (distinct or identical). Show that ~ is an equivalence relation.

10. Prove that in the set N × N, the relation R defined by (a, b) R (c, d ) ad = bc is an equivalence relation.

1.Prove that any graph (not necessarily a tree) withvvertices andeedges that satisfiesv>e+1v>e+1will NOT be connected.

1.    We define a forest to be a graph with no cycles.

a)    Explain why this is a good name. That is, explain why a forest is a union of trees.

b)   Suppose FF is a forest consisting of mm trees and v vertices. How many edges does FF have? Explain.

c)    Prove that any graph GG with v vertices and e edges that satisfies v<e+1 must contain a cycle (i.e., not be a forest).

VI.

Rewrite the following conditional statement in “If p, then q.” form.

1. I will be able to receive my credential provided that Education 147 is offered in the spring

semester.

2. It is an ankylosaur only if it is quadrupedal.

3. Every odd prime number is greater than 2.

4. We can get a dog only if we install a fence around the backyard.

5. The triangle is 30º- 60º-90º triangle, if the length of the hypotenuse is twice the length of

the shorter leg.

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