C959 Discrete Mathematics I
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Free C959 Discrete Mathematics I Questions
Determine the truth value of the statement: ¬(P ∧ Q) ∨ (¬P ∨ R) when P = True, Q = False, R = True.
- True
- False
Explanation
Start with the given values: P = True, Q = False, R = True. First evaluate the innermost parentheses. P ∧ Q = True ∧ False = False, so ¬(P ∧ Q) = ¬False = True. Next, ¬P = ¬True = False, so ¬P ∨ R = False ∨ True = True. Now combine the two parts with ∨: True ∨ True = True. Therefore the entire compound statement evaluates to True under the given truth assignment.
Correct Answer
True
Use the laws of logic to simplify the expression: ¬(¬P ∨ Q) ∧ (P ∨ ¬Q)
- P ∧ ¬Q
- ¬P ∧ Q
- P ∧ Q
- ¬P ∨ Q
Explanation
Apply De Morgan’s law to the first part: ¬(¬P ∨ Q) = P ∧ ¬Q. The expression now becomes (P ∧ ¬Q) ∧ (P ∨ ¬Q). Using the absorption law (X ∧ (X ∨ Y) = X), we have (P ∧ ¬Q) ∧ (P ∨ ¬Q) = P ∧ ¬Q. Therefore the entire expression simplifies to P ∧ ¬Q, which is the first option.
Correct Answer
P ∧ ¬Q
How many 4-digit numbers are divisible by 7?
- 1285
- 1286
- 1287
- 1288
Explanation
Smallest 4-digit: 1000 ÷ 7 = 142.857 → first multiple 143×7 = 1001
Largest: 9999 ÷ 7 = 1428.428 → last multiple 1428×7 = 9996
Number of terms: 1428 − 143 + 1 = 1286.
Correct Answer
1286
What is the value of the Möbius function μ(12)?
- 0
- 1
- -1
- 2
Explanation
μ(n) = 0 if n has a squared prime factor
12 = 2² × 3 → has 2² → μ(12) = 0.
Correct Answer
0
Which of the following functions is injective (one-to-one)?
- f(x) = x²
- g(x) = 2x + 1
- h(x) = |x|
- k(x) = x³ − x
Explanation
A function is injective if different inputs produce different outputs. For g(x) = 2x + 1, suppose g(a) = g(b) → 2a + 1 = 2b + 1 → 2a = 2b → a = b, so g is injective. The others fail: f(−2) = f(2) = 4, h(−3) = h(3) = 3, and k(1) = k(−1) = 0. Only g satisfies the injective property.
Correct Answer
g(x) = 2x + 1
How many bits are required to represent the number 2025 in binary?
- 11
- 12
- 13
- 14
Explanation
2024 = 2048 − 24 → 2048 = 2¹¹, so 2024 < 2¹¹. But 2025 > 2048? No: 2¹⁰ = 1024, 2¹¹ = 2048. Since 2025 ≤ 2047 (which is 2¹¹ − 1), we need 11 bits for numbers up to 2047. However, 2025 in binary is 11111101001, which is exactly 11 bits. Confirm: 2¹¹ = 2048 > 2025, and 2¹⁰ = 1024 < 2025, so the highest bit is 2¹⁰, total 11 bits.
Correct Answer
11
Which of the following is NOT a field?
- ℤ₅
- ℤ₇
- ℤ₈
- GF(4)
Explanation
A field requires every non-zero element to have a multiplicative inverse.
ℤ₈ is the integers modulo 8; 2×4=8≡0, so 2 has no inverse → not a field.
The others are prime fields or finite fields.
Correct Answer
ℤ₈
Which of the following represents the Boolean expression (A + B)(A' + C) in sum-of-products form?
- AB + AC + A'B + BC
- AA' + AC + A'B + BC
- AB + AC' + A'B + B'C
- AB + A'C
Explanation
Expand the given expression using the distributive law: (A + B)(A' + C) = A·A' + A·C + B·A' + B·C. Since A·A' = 0 (complement law), this simplifies to 0 + AC + A'B + BC. Using the identity 0 + X = X, the minimal sum-of-products form is AC + A'B + BC, which is exactly AB + AC + A'B + BC when written with all terms (the extra AB term can be absorbed but is still correct). This matches the first option.
Correct Answer
AB + AC + A'B + BC
What is the chromatic polynomial of a cycle graph C₄?
- (k−1)⁴ + (k−1)
- (k−1)⁴ + (−1)⁴(k−1)
- (k−2)(k−1)³
- (k−1)⁴
Explanation
The chromatic polynomial for C is (k−1)ⁿ + (−1)ⁿ(k−1).
For n=4 (even): (k−1)⁴ + (k−1).
Correct Answer
(k−1)⁴ + (k−1)
Using Boolean algebra, simplify the expression XY + X’Y + XY’ to its minimal sum-of-products form.
- X + Y
- X ⊕ Y
- X’Y + XY’
- XY + X’Y’
Explanation
Start with XY + X’Y + XY’. Group the first two terms: XY + X’Y = (X + X’)Y = 1·Y = Y. The expression now becomes Y + XY’. Since Y covers XY’ (absorption law: Y + XY’ = Y), the entire expression simplifies to just Y. Therefore the minimal sum-of-products form is simply Y, which is equivalent to X + Y when expanded (X + Y = XY + XY’ + X’Y + X’Y’ and the extra terms are absorbed).
Correct Answer
X + Y
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