Discrete Mathematics Answers

1a) Use set-builder notation to prove that

A-(B ∩C)=(A-B)∪(A-C)

b) Let E={ 1,23,4,5,6,7,8,9,10} and ordering of element in increasing  order that is a= i; what bit strings represent the subsets of integers not exceeding  5 in E

3a)Let A={1,2,3},determine  all partitions of A

1. Classify and compare the different types of antenna using ah venn diagram. Discuss each type in each petal of the diagram.

Construct a K-map for F(x, y, z) = xz + yz + xyz. Use this K-map to find the implicants, prime

implicants, and essential prime implicants of F(x, y, z).

Let A = {2, 4} and B = {2, 4, 6} and define a relation R for A to B as follows:

Question: Which ordered pairs belongs to relation R?

A gumball machine contains 300 grape flavored balls, 400 cherry flavored balls, and 500 lemon flavored balls. What is the probability of getting 1 grape ball, 1 cherry ball, and 1 lemon ball if each ball was removed and then replaced before choosing the next from the machine?

Show that if n is a positive integer with n ≥ 3, then

C(n, n − 2) = ((3n − 1)C(n, 3))/4

What are the values of the following expressions?

• ∑_(i=3)^6▒ (i^2+6)
• ∏_(j=1)^5▒ (j/2)
• 4!
• 0!
• (6¦2)
• C(6,2)

Answer these questions for the poset ({3, 5, 9, 15, 24, 45}, |).

a) Find the maximal elements.

b) Find the minimal elements.

c) Is there a greatest element?

d) Is there a least element?

e) Find all upper bounds of {3, 5}.

f) Find the least upper bound of {3, 5}, if it exists.

g) Find all lower bounds of {15, 45}.

h) Find the greatest lower bound of {15, 45}, if it exists.

Draw the directed graphs representing each of the relations on {1, 2, 3, 4}

a) {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}

b) {(1, 1), (1, 4), (2, 2), (3, 3), (4, 1)}

c) {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)}

d) {(2, 4), (3, 1), (3, 2), (3, 4)}

Represent each of these relations on {1, 2, 3} with a matrix (with the elements

of this set listed in increasing order).

a) {(1, 1), (1, 2), (1, 3)}

b) {(1, 2), (2, 1), (2, 2), (3, 3)}

c) {(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)}

d) {(1, 3), (3, 1)}