# COMP-047 Discrete mathematics Exam

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COMP-047 Exam #1

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1) (6 pts) Show that (p ?q) ? ( ~q ? ~p) a tautology.

2) (4 pts) If A = {1,2 3,4} and B = {a,b} what is the cardinality of the powerset of AXB?

3) (7 pts) Give a direct proof of the following: If x,y are integers and y is odd, then 2x + y + 1 is even

4) (5 pts) Consider the set A = {1,2 3} and B = { a,b,c,d} and the function f: A ? B. What is the cardinality of f?

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Attached.

Running head: DISCRETE MATHEMATICS EXAM

1

Discrete mathematics

Name of Student

Name of Institution

DISCRETE MATHEMATICS EXAM

1. Show that (p →q) ↔ (~q → ~p) a tautology. (6 points)

De Morgan’s laws states that intersection of two complement sets is similar union of two sets

complement and vice versa.

Therefore,

Not A or nor B is the same to not (A and B)

Not A and Not B is the same as not (A or B)

In V=Boolean algebra and set theory it is written as

Where

A and B are sets

A compliment A

∪ is the union.

∩ is the intersection

Rules written in formal language

And

Where

P and Q are propositions,

(ᴧ) (AND) is the conjunction logic operator,

(¬) (NOT) is the negation logic operator,

(˅) (OR) is the disjunction logic operator,

2

DISCRETE MATHEMATICS EXAM

3

2. If A = {1,2 3,4} and B = {a,b} what is the cardinality of the powerset of AXB?(4 pts)

Cardinally refers to the number of elements in a finite set and power set of A or P (A). Hence,

cardinality of P (A) refers to the number of subsets of A.

Here, A=0,1,2,3,4,5,6,7,8,9,10

|A|

I.e. cardinality of A is 11.

If A

Is a finite set with |A| = n elements, then the number of subsets of A is=2n.

Again, |P (A)|

= number of subsets of A=2n

Putting n=11

, we get that |P (A)| is 211

3. Give a direct proof of the following: If x,y are integers and y is odd, then 2x + y + 1 is

even (7 pts)

x,y are integers, and y is odd

2x + y + 1 is even

x and y are integers

and y is odd, set y be 2n+ 1

x can either be odd or even

But

2x+y+1= ˃ here 2x is even since x is multiplied by 2

DISCRETE MATHEMATICS EXAM

4

y is odd, then y+1 is even

2𝑋

𝑒𝑣𝑒𝑛

+

𝑦+1

𝑒𝑣𝑒𝑛

=˃Adding two even numbers is again an even

Hence proved

4. Consider the set A = {1,2 3} and B = { a,b,c,d} and the function f: A → B. What is the

cardinality of f?(5 pts.)

A = {1, 2, 3}

B = {a, b, c, d}

f: A -˃ B

Cardinality of f = cardinality of A

= A (number of elements A)

=3

5. Rewrite the following using universal quantification. (4 pts)

Universally Qualified Statement

We assume the exp...

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