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Module-1 of Model Paper-1 (Explained with solutions)
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Introduction
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1a] Show that the compound proposition ——— for primitive statements p, q, r is logically equivalence.
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1b] Establish the validity of the following argument using the Rules of Inference: {p ∧ (p → q) ∧ (s ∨ r) ∧ (r →∼ q)} ⟶ (s ∨ t)
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1c] For the universe of all integers, let p(x), q(x), r(x), s(x) and t(x) denote the following open statements: p(x): x>0, q(x): x is even, r(x): x is a perfect square, s(x): x is divisible by 3, t(x): x is divisible by 7. Write the following statements in symbolic form: i) At least one integer is even. ii) There exists a positive integer that is even. iii) If x is even, then x is not divisible by 3. iv) No even integer is divisible by 7. v) There exists even integer divisible by 3.
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2a] Define a tautology. Prove that, for any propositions p, q, r the compound propositions, {(p→q)∧(q→r)}→{(p→r)} is tautology.
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2b] Prove the following using laws of logic: p→(q→r)⇔(p∧q)→r.
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2c] Give i) direct proof ii) indirect proof iii) proof by contradiction for the following statement: “if n is an odd integer then n+9 is an even integer”.
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Module-2 of Model Paper-1 (Explained with solutions)
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3a] Prove that 1 2 + 32 + 52 + ⋯ . +(2n − 1) 2 = n(2n+1)(2n−1) 3 by Mathematical Induction. 3b]Let a0 = 1, a1 = 2, a2 = 3 and an = an−1 + an−2 + an−3 for n≥3. Prove that an ≤ 3n ∀ n ∈ z+. 3c] Find the number of ways of arrangement of the letters of the word ‘TALLAHASSEE’ which have no adjacent A’s.
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4a] Determine the coefficient of xyz 2 in the expansion of (2x − y − z) 4 . 4b] In how many ways one can distribute 8 identical marbles in 4 distinct containers so that i) no container is empty ii) the fourth container has an odd number of marbles in it. 4c] How many positive integers n can we form using the digits 3,4,4,5,5,6,7 if we want n to exceed 5,000,000?
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Module-3 of Model Paper-1 (Explained with solutions)
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5a] Let f: R → R be defined by, f(x) = { 3x − 5 if x > 0 1 − 3x if x ≤ 0 . Find f −1(0), f−1(1), f−1(−1), f−1(3), f−1(6), f −1([−6, 5]) and f −1([−5, 5]) 5b] State Pigeon hole principle. Prove that in any set of 29 persons; at least 5 persons have been born on the same day of the week. 5c] Let A={1,2,3,4,6} and ‘R’ be a relation on ‘A’ defined by aRb if and only if “a is multiple of b” represent the relation ‘R’ as a matrix, draw the diagraph and relation R.
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6a] If f:A→B, g:B→C, h:C→D are three functions, then Prove that h∘(g∘f)=(h∘g)∘f. 6b] Show that if n+1 numbers are chosen from 1 to 2n then at least one pair add to 2n+1. 6c] Draw the Hasse diagram representing the positive divisors of 72.
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Module-4 of Model Paper-1 (Explained with solutions)
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7a] In how many ways the 26 letters of English alphabet are permuted so that none of the pattern’s CAR, DOG, PUN or BYTE occurs 7b] Define Derangement. In how many ways can each of 10 people select a left glove and a right glove out of a total of 10 pairs of gloves so that no person selects a matching pair of gloves? 7c] Solve the recurrence relation: Cn = 3Cn−1 − 2Cn−2, for n ≥ 2, given C1 = 5, C2 = 3.
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8a] In how many ways one can arrange the letters of the word CORRESPONDENTS so that there are i) exactly 2 pairs of consecutive identical letters? ii) at least 3 pairs of consecutive identical letters? iii) no pair of consecutive identical letters? 8b] Find the rook polynomial for the chess board as shown in the figure 8c] Solve the recurrence relation an+2 − 3an+1 + 2an = 0, a0 = 1, a1 = 6.
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Module-5 of Model Paper-1 (Explained with solutions)
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9a] If H, K are subgroups of a group G, prove that H ∩ K is also a subgroup of G. Is H ∪ K a subgroup of G. 9b] Define Klein 4 group. Verify A = {1, 3, 5, 7} is a klein 4 group. 9c] State and prove Lagrange’s Theorem.
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10a] Show that i) the identity of G is unique. ii) the inverse of each element of G is Unique. 10b] Show that (A, ∙) is an abelian group where A = {a ∈ Q|a ≠ −1} and for any a, b ∈ A, a . b = a + b + ab. 10c] Let G = S4, for α = (1 2 3 2 3 4 4 1 ), find the subgroup H = 〈α〉. Determine the left cosets of H in G.
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DISCRETE MATHEMATICAL STRUCTURES || BCS405A || Model Question Paper-1 with solutions (EXPLAINED)|| 2022 scheme || Video course with study materials
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