Discrete Mathematics — Flashcards
Tamil Nadu Board · Class 12 · Mathematics
27 flashcards for Discrete Mathematics (Tamil Nadu Board Class 12 Mathematics) to test yourself on key terms and facts.
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Is subtraction (-) a binary operation on the set of natural numbers ℕ = {1, 2, 3, ...}? Justify your answer.
Answer
No, subtraction is NOT a binary operation on ℕ. Reason: For (a, b) ∈ ℕ × ℕ, the result (a - b) may not be in ℕ. Counterexample: (3, 5) ∈ ℕ × ℕ, but 3 - 5 = -2 ∉ ℕ. Since the output is not always in ℕ,…
Verify if the operation a * b = a + b - ab is commutative on ℤ. Show working.
Answer
Check if a * b = b * a for all a, b ∈ ℤ. Left side: a * b = a + b - ab. Right side: b * a = b + a - ba = b + a - ab (since multiplication is commutative). Therefore a * b = a + b - ab = b + a - ab = b…
For the operation a * b = a + b - 5 on ℤ, find the identity element e such that a * e = e * a = a for all a ∈ ℤ.
Answer
We need a * e = a for all a ∈ ℤ. Substituting: a + e - 5 = a. Simplifying: e - 5 = 0, so e = 5. Verify: a * 5 = a + 5 - 5 = a ✓. Also check: 5 * a = 5 + a - 5 = a ✓. The identity element is e = 5. Thi…
For the operation m * n = m + n - mn on ℤ, determine if an inverse element exists for every element. If not, explain why.
Answer
To find inverse of m, we need m' such that m * m' = 0 (the identity from previous analysis). So: m + m' - mm' = 0. Rearranging: m' - mm' = -m, so m'(1 - m) = -m, thus m' = m/(m-1). For m = 1: m' is un…
Determine if a * b = min(a, b) is associative on the set A = {1, 2, 3, 4, 5}. Use an example.
Answer
Check if (a * b) * c = a * (b * c). Let's test with a = 3, b = 5, c = 2. Left side: (3 * 5) * 2 = min(3, 5) * 2 = 3 * 2 = min(3, 2) = 2. Right side: 3 * (5 * 2) = 3 * min(5, 2) = 3 * 2 = min(3, 2) = 2…
Calculate 5 +₇ 6 and 3 ×₇ 5 using modular arithmetic (mod 7).
Answer
For addition modulo 7: 5 +₇ 6 means find the remainder of (5 + 6) ÷ 7. 5 + 6 = 11. 11 ÷ 7 = 1 remainder 4. So 5 +₇ 6 = 4. For multiplication modulo 7: 3 ×₇ 5 means find the remainder of (3 × 5) ÷ 7. 3…
Verify that 27 ≡ 3 (mod 8). What does this notation mean mathematically?
Answer
The notation 27 ≡ 3 (mod 8) means 27 and 3 are congruent modulo 8. Mathematically: (27 - 3) is divisible by 8. Check: 27 - 3 = 24 = 8 × 3 ✓. Alternatively: 27 ÷ 8 = 3 remainder 3, and 3 ÷ 8 = 0 remain…
Using the addition modulo 5 table, find the inverse of 3 in ℤ₅ = {0, 1, 2, 3, 4}.
Answer
The inverse of 3 in ℤ₅ under addition is the element x such that 3 +₅ x = 0 (the identity). Using the modulo 5 addition table: Row 3 is [3, 4, 0, 1, 2]. Find where 0 appears: it's in the column under …
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