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 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?

(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.

Prove that the conditional proposition p->q and its contrapositive ~q ->~p are logically equivalent using the truth table.

Prove by mathematical induction the formula (1^3+2^3+3^3+4^3+....n^3)=(n^2(n+1)^2)/4