Discrete Mathematics Answers

Prove by mathematical induction that: Where "E" is the summation icon.

n i
E E j = 1/6n(n+1)(n+2)
i = 1 j=1

Let C = {1, 4, 5} and D = {2, 7}.

• List the elements of C × D.
• If there are 140 elements in D × A, how many elements are in A?
• Give an example of a relation from D to C that is not a function. Draw a graph of the
relation. Explain why it is not a function.
• Give an example of a relation from C to D that is a function. Draw an arrow diagram for the relation

Let A = {1, 2, 3, 4} and let R be a relation on A such that R = {(1, 1),(2, 2),(3, 3),(4, 4),(1, 2),(2, 3),(1, 3)}

Is R transitive? Symmetric? Reflexive?

Let A = {1, 2, 3, 4} and let R be a relation on A such that R = {(1, 1),(2, 2),(3, 3),(4, 4),(1, 2),(2, 3),(3, 2),(2, 1)}

Is R transitive? Symmetric? Reflexive?

Draw the graph represented by the adjacency matrix . (001 010 100)

(p∧q)=(q∧p)
and
(p∨q)=(q∨p)
implies an..............

Answers Options
a. Associative laws
b. Distributive Laws
c. Commutative Laws
d. Idempotent Laws

b) Given that p and q are propositions construct the truth table of; (4 Marks)
i. p->q
ii. p<->q

6.which of the following statement is true

a.(p∨q)∧(p∨r)=p∨(q∧r)

b.(p∧q)∧(p∨r)=p∨(q∧r)

c.(p∧q)∨(p∨r)=p∨(q∧r)

d.∼(p∨q)=∼(p∧∼q)

7. ____ reads “the goods are standard if and only if the goods are expensive”

a.p↔q

b.∼p∧q

c.∼∼q

d.∼(∼p∧∼q)

8.______is a rectangular array of numbers with reference to specific rules governing the array
a.Transformation
b.Function
c.Relation
d.Matrix

9. A matrix in which its transpose is equal to itself is called____
a.power matrix of A
b.symmetric matrix
c.identity matrix
d.transpose matrix

10. (p∨q)∨r=p∨(q∨r) and (p∧q)∧r=p∧(q∧r)
implies an ___
a.Associative laws
b.Idempotent Laws
c.Commutative Laws
d.Distributive Laws

According to information obtained from mathematics department regarding three mathematics units done by 100 students, those who are doing calculus are 45, those doing discrete are 49 and those doing statistics are 38. Those doing calculus and discrete are 17, those doing calculus and statistics are 12 and those doing the three units are 5. Use Venn diagram to find the number of students doing discrete and statistics but not calculus.

Recall that a real number x is rational if x = p/q for integers p, q with q = ̸= 0.

Prove that if x is rational then 1/(2x+1) is rational. Then prove that if 1/(2x+1) is rational then x is rational.