Discrete Mathematics Answers

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)}

Show that the relation R = ∅ on the empty set S = ∅ is

reflexive, symmetric, and transitive.

Show that the relation R = ∅ on a nonempty set S is sym-

metric and transitive, but not reflexive.

Suppose a recurrence relation

an=7an−1−12an−2

where a1=16 and a2=52

can be represented in explicit formula, either as:

Formula 1:

an=pxn+qnxn

                      or  

Formula 2:

an=pxn+qyn

where 

x

and

y

are roots of the characteristic equation.

**If the explicit formula is in the form of Formula 2, consider p < q.

Determine

p and q

. In how many ways a relation can be represented? State two different

examples to represent each of them.

In how many ways a relation can be represented? State two different

examples to represent each of them.

What is nested Quantifier? Is order important for nested quantifier?

Explain your answer with appropriate example.