Discrete Mathematics Answers

Exhibit a cycle of length 9 in this graph

1.Find a possible statistical problem inside your home relating to a discrete random variable lesson, explain in a paragraph form…..(2-3 sentence will do)

2.Find the possible random variable

3.Create a probability distribution

4.Compute for the mean variance and standard deviation

5.Interpret result

6.Conclusion

• Prove that 7n−17n−1 is a multiple of 6 for all n∈N.

• Use induction to prove for all n∈Nn∈N that n∑k=02k=2n+1−1.

• Prove that (ab)n=aⁿbⁿ(ab)n=anbn is true for every natural number n

Qu 01: Model the following situations as (possibly weighted, possibly directed) graphs. Draw each graph, and give the corresponding adjacency matrices.

(i) Rokey and Jane are friends. Rokey is also friends with Kevin and Ferdy. Jane ,Kevin and kanny are all friends of each other.

Let R be the relation on Z

 (the set of integers) defined by 

(x, y)  R iff x

2 + y2 = 2k for some integers k  0.

Question 15

R is not antisymmetric. Which of the following ordered pairs can be used together in a 

counterexample to prove that R is not antisymmetric? (Remember that R is defined on Z



.)

1. (–1, 1) & (1, –1)

2. (5, 9) & (13, 15)

3. (8, 7) & (7, 8)

4. (3, 1) & (1, 3)

Let R be the relation on Z

 (the set of integers) defined by 

(x, y)  R iff x

2 + y2 = 2k for some integers k  0.

Answer questions 13 to 15 by using the given relation R.

Question 13

Which one of the following is an ordered pair in R?

1. (1, 0)

2. (2, 9)

3. (3, 8)

4. (5, 7)

Question 14

R is symmetric. Which one of the following is a valid proof showing that R is symmetric?

1. Let x, y  Z be given.

Suppose (x, y)  R

then x

2 + y2 = 2k for some k  0.

ie y

2 + x2 = 2k for some k  0.

thus (x, y)  R.

2. Let x, y  Z be given.

Suppose (x, y)  R

then x

2 + y2 = 2k for some k  0.

ie y

2 + x2 = 2k for some k  0.

thus (y, x)  R.

3. Let x, y  Z

 be given.

Suppose (x, y)  R

then x

2 + y2 = 2k for some k  0.

thus (y, x)  R.

4. Let x, y  Z be given.

Suppose (x, x)  R

then x

2 + y2 = 2k for some k  0.

ie y

2 + x2 = 2k for some k  0.

thus (y, y)  R.

Mary Ann pays a monthly fee for her cell phone package which includes 700 minutes. She gets Billed an additional charge for every minute she uses the phone past the 700 minutes. During her first month, Mary Ann used 26 additional minutes and her bill was $37.79. During her second month, Mary Ann used 38 additional minutes and her bill was $41.39.

i. Write a function that represents the total monthly cost C(x) of Mary Ann’s cell phone package, where x is the number of additional minutes used.

ii. Find the inverse function

iii. What do x and C-1(x) represent in the context of the inverse function?

iv. How many additional minutes did Mary Ann use if her bill for her third month was $48.89?

 Define intersection of two Fuzzy set with example.