MTH202 Midterm Solved MCQs and Subjective with References by Sadia

Discrete Mathematics · MTH 202 · Semester Fall 2004 · Total Marks: 40 Time: 60 min · Instructions · question:

1. The duration of this examination is 60 minutes.

2. This examination is closed book, closed notes, closed

neighbors.

3. Answer all questions.

there is something wrong with any question then made the best assumptions which you think and

then answer that

question.

5. Calculator is allowed

Q1. Marks : 2

If R ={ (a, a),(b, b),(c, c)} is a relation on the set A={a, b, c} Then R is

a) Symmetric only. b) Symmetric and reflexive only. c) Reflexive only.

d) Equivalence

relation

Q2. The negation of the implication “If P is a square then P is a rectangle” is

1. If P is not a square then P is not a rectangle
2. P is not a square and P is a rectangle
3. P is a square and P is not a rectangle.
4. None of the above

Q3. Identify the false statement

1. 0 ∈∅
2. { ∅
3. If A and B are two sets A B and B A then A = B .
4. Two sets are disjoint if their intersection is empty set.
5. A ∪A c = U

Q4. Let A be a set containing 3 elements then the total number of relations from A to A

is 1. 2*9

2. 29

3. nn 4. 2n

Q5. Let A = {1,2,3} and B = {2,3,4,5} then

1. A = B.
2. A is a subset of B.
3. A is improper subset of B.
4. Both 2 and 3.

Q6. Marks : 5

Construct truth table for the following compound proposition p q p q

solution: · p · q · p · q

p ∧q

p

q

↔ p ∧q T T T T T T F F F T F T T F F F F T F F

1 marks for column 1 1 marks for column 2 1 marks for column 3 1 marks for column 4 1 marks for column 5

Q7. Marks : 5

What are the contra positive , the inverse , and the converse of the implication “If you have flu then you will miss the final examination”

Solution:

Let p = you have flu Q = you will miss the final examination

Contra positive

~ q → ~p

If you will not miss the final examination then you have no flu.

Inverse

~ p → ~q

If you have no flu then you will not miss the final examination

Converse · q · p

If you will miss the final examination then you have flu.

2 marks for contra positive 2 marks for the inverse 1 marks for the converse

Q8. Determine whether the

relation R on the set of integers is transitive, where x, y ∈R if and only if 2|(x – y)

Solution:

To show that R is reflexive, it is necessary to show that For all x, y, z ∈Z if x R y, y R z then x R z

By definition of R this means that

For all x, y, z ∈ Z, if 2|(x – y), 2|(y – z) then 2| (x – z)

Now by definition of “divides”

2|(x – y) x – y = 2k for some integer k

Now by definition of “divides”

2|(y – z) y – z = 2m for some integer m (x – y) +( y – z) = 2k + 2m x – z = 2 (k + m)

Since k and m are integers then k + m is also an integer Let k + m = s for some integer s

x – z = 2s This implies that 2| (x – z) It follows that R is Transitive

2 marks for the definition 2 marks for the calculations 1 marks for the required result