Math and Science Program
Asian University for Women
Midterm, Fall-2021 (Set-6)
Course Code: MATH 3006, Title: Discrete Mathematics
Total Marks: 40
Total Time: 1 Hour
[Note: Answer any 5 out of 6 questions. Each question carries equal mark]
1. Construct a truth table for each of the following compound propositions:
a) (p ∨ q) ⇾ (p ⨁ q)
b) (p ⨁ q) ∨ (⌝ p ⨁ q)
2. a) Determine the truth value of each of these statements if the domain consists of all
integers.
i) ∃n(n = −n)
ii) ∀n(3n ≤ 5n)
b) Let P(x, y) be the statement “x has sent an email to y,” where the domain for both x
and y consists of all students in your class. Express each of these quantifications in
English.
i) ∀x∃yP(x, y)
ii) ∃y∀xP(x, y)
iii) ∀x∀yP(x, y)
3. Prove the following statements using any of the methods of direct proof, proof by
contraposition, or proof by contradiction.
a) “If n is an odd integer, then n2 is odd.”
b) √8 is an irrational number.
4. a) Determine the Truth value of the following if x and y are integers:
∀x (x≠0) → ∃y (xy =-2)
Give an example for each in favor of your Truth values.
b) For any integer n, “If n2 is odd, then n is odd”, proof it by contradiction.
5. a) Find the bitwise OR, bitwise AND, and bitwise XOR of the bit strings 01 0011 0110
and 11 0001 0101.
b) Prove that n2 + 3≥ 2n when n is a positive integer with 1 ≤ n ≤ 4.
6. a) Draw the Venn diagrams for each of these combinations of the sets A, B, and C.
i) (A ∩ B) ∪ (A ∩ C)
ii) (A ∩ B) ∪ (A ∩ C)c
b) Suppose that the universal set is U = {1,2,3,4,5,6,7,8,9,10}. Express each of these sets
with bit strings or find the sets specified by the bit strings.
i) {2, 4, 5, 6, 9}
ii) 11 1010 1011