Discrete Mathematics

Discrete Mathematics One Mark Questions PREPARED BY: R.RAJENDRAN. M.A., M. Sc., M. Ed., K.C.SANKARALINGA NADAR HR. SEC. SCHOOL, CHENNAI-21

Choose the Correct Answer • Which of the following statements are statements?(i) May God bless you (ii) Rose is a flower(iii) Milk is white (iv) 1 is a prime number(a) (i), (ii), (iii) (b) (i), (ii), (iv)(c) (i), (iii), (iv) (d) (ii), (iii), (iv)

Choose The Correct Answer • If a compound statement is made up of three simple statements, then the number of rows in the truth table is(a) 8 (b) 6(c) 4 (d) 2 • The number of rows in the truth table of [p٨(q)] is(a) 2 (b) 4 (c) 6 (d) 8

Choose The Correct Answer • If p is T and q is F, then which of the following have the truth value T?(i) p ٧ q (ii)  p ٧ q (iii) p ٧ q (iv) p ٨ (q)(a) (i), (ii), (iii) (b) (i), (ii), (iv)(c) (i), (iii), (iv) (d) (ii), (iii), (iv) • The conditional statement pq is equivalent to(a) p ٧q (b) p ٧ q (c)  p ٧q (d) p ٨q

Choose The Correct Answer • Which of the following is a tautology?(a) p ٧q (b) p ٨ q (c) p ٧ p (d) p ٨ p • Which of the following is a contradiction?(a) p ٧q (b) p ٨ q (c) p ٧ p (d) p ٨ p

Choose The Correct Answer • pq is equivalent to(a) p q (b) q  p (c) (p q)٧ (q  p) (d) (p q)٨ (q  p) • Which of the following is not a binary operation on R?(a) a * b = ab (b) a * b = a – b (c) a * b = (d) a * b =

Choose The Correct Answer • A monoid become a group if it also satisfies the(a) closure axiom (b) associative axiom (c) identity axiom (d) inverse axiom • Which of the following is not a group?(a) (Zn, +n) (b) (Z, +) (c) (Z, .) (d) (R, +)

Choose The Correct Answer • In the set of integers with operation * defined by a * b = a + b – ab, the value of 3 * (4 * 5) is(a) 25 (b) 15 (c) 10 (d) 5 • The order of [7] in (Z9, +9) is(a) 9 (b) 6 (c) 3 (d) 1

Choose The Correct Answer • In the multiplicative group of cube root of unity, the order of 2 is is(a) 4 (b) 3 (c) 2 (d) 1 • The value of [3] +11 ([5] +11 [6]) is(a) [0] (b) [1] (c) [2] (d) [3]

Choose The Correct Answer • In the set of real numbers R, an operation * is defined by a * b = , then the value of (3 * 4) * 5 is(a) 5 (b) 52(c) 25 (d) 50 • The order of – i in the multiplicative group of 4th roots of unity is(a) 4 (b) 3 (c) 2 (d) 1

Choose The Correct Answer • Which of the following is correct?(a) An element of a group can have more than one inverse.(b) If every element of a group is its own inverse, then the group is abelian.(c) The set of all 2  2 real matrices forms a group under matrix multiplication(d) (a * b)–1 = a–1 * b–1 for all a, b G

Choose The Correct Answer • In the multiplicative group of nth roots of unity, the inverse of k is (k < n)(a) 1/k (b) – 1 (c) n – k (d) n/k • In the set of integers under the operation * defined by a*b = a + b – 1, the identity element is(a) 0 (b) 1 (c) a (d) b

Choose The Correct Answer • The number of possible binary operation defined on a set with 5elements is(a) 25 (b) 52 (c) 10 (d) 5 • The inverse of – 1 in the group (Z, +) is(a) 0 (b) – 2 (c) 2 (d) 1

Choose The Correct Answer • Which one of the following is false?(a) (N, +) is a semi group (b) (N, .) is a monoid(c) (Z, *) is a group (d) the set of all even integers with usual addition is a group • The value of [3] .5 [4] is(a) [0] (b) [4] (c) [1] (d) [2]

Choose The Correct Answer • If a binary operation * is defined by a * b = 3a – b, then the value of (2 * 3) * 4 is (a) 2(b) 3(c) 4 (d) 5 • (N, .) is a (a) group (b) semi group (c) monoid (d) abelian group

Choose The Correct Answer • If (G, *) is a group and a, bG, then the solution of the equation a * x = b is(a) a * b–1(b) a–1 * b(c) b * a–1 (d) b–1 * a • The value of [2] +8 [4] +8 [6] is(a) [0] (b) [2] (c) [3] (d) [4]

Choose The Correct Answer • In the multiplication group of matrices of the formxR – {0}, the identity element is(a) (b) (c) (d) • Which one of the following is not a binary operation on the set of integers(a) addition (b) subtraction(c) a * b = ab/3 (d) a * b = a + b + 1

Choose The Correct Answer • Which of the following are statements?(1) Chennai is capital of Tamil nadu (2) The earth is a planet(3) Rose is a flower (4) Every triangle is an isosceles triangle(a) all (b) 1 and 3 (c) 2 and 3 (d) 4 only

Choose The Correct Answer • Which of the following are not statements?(1) Three plus four is eight (2) The sun is a planet(3) Switch on the light (4) Where are you going?(a) 1 and 3 (b) 2 and 3 (c) 3 and 4 (d) 4 only

Choose The Correct Answer • The truth values of the following statements are(1) Ooty is in Tamil nadu and 3 + 4 = 8 (2) Ooty is in Tamil nadu and 3 + 4 = 7(3) Ooty is in Kerala and 3 + 4 = 7 (4) Ooty is in Kerala and 3 + 4 = 8(a) FTFF (b) FFFT (c) TTFF (d) TFTF

Choose The Correct Answer • The truth values of the following statements are(1) Chennai is in India or 2 is an integer (2) Chennai is in India or 2 is an irrational number(3) Chennai is in China or 2 is an integer (4) Chennai is in China or 2 is an irrational number(a) TFTF (b) TFFT (c) FTFT (d) TTFT

Choose The Correct Answer • Which of the following are not statements?(1) All natural numbers are integers (2) A square has five sides(3) The sky is blue (4) How are you?(a) 4 only (b) 1 and 4 (c) 1, 2 and 3 (d) 3 and 4

Choose The Correct Answer • Which of the following are statements?(1) 7 + 2 < 10 (2) The set of rational numbers is finite(3) How beautiful you are! (4) Wish you all success(a) 3 and 4 (b) 1 and 2 (c) 1 and 3 (d) 2 and 4

Choose The Correct Answer • The truth values of the following statements are(1) All the sides of a rhombus are equal in length (2) 1 + 19 is an irrational number(3) Milk is white (4) The number 30 has four prime factors(a) TTTF (b) TTTT (c) TFTF (d) FTTT

Choose The Correct Answer • The truth values of the following statements are(1) Paris is in France (2) sin x is an even function(3) Every square matrix is non-singular (4) Jupiter is a planet(a) TFFT (b) FTFT (c) FTTF (d) FFTT

Choose The Correct Answer • Let p be “Kamala is going to school” and q be “There are twenty students in the class”. “Kamala is not going to school or there are twenty students in the class” stands for(a) p  q (b) p  q (c) ~ p (d) ~ p  q

Choose The Correct Answer • If p stands for the statement “Sita likes reading” and q stands for the statement “Sita likes playing”. “ Sita likes neither reading nor playing” stands for(a) ~ p  ~ q (b) p  ~ q (c) ~ p  q (d) p  q

Choose The Correct Answer • If p is true and q is unknown then(a) ~ p is true (b) p  (~ p) is false (c) p  (~ p) is true (d) p  q is true • If p is true and q is false then which of the following is not true?(a) ~ p is true (b) pq is true (c) p q is false (d) pq is true

Choose The Correct Answer • Which of the following is not true?(a) Negative of a negation of a statement is the statement itself(b) If the last column of its truth table contains only T then it is tautology(c) If the last column of its truth table contains only F then it is contradiction(d) If p and q are any two statements then pq is a tautology

Choose The Correct Answer • Which of the following are binary operations on R?(1) a * b = min{a, b} (2) a * b = max{a, b} (3) a * b = a (4) a * b = b(a) all (b) 1, 2 and 3 (c) 2, 3 and 4 (d) 3 and 4

Choose The Correct Answer • ‘+’ is not a binary operation on(a) N (b) Z (c) C (d) Q – {0} • ‘–’ is a binary operation on(a) N (b) Q – {0} (c) R – {0} (d) Z

Choose The Correct Answer • ‘’ is a binary operation on(a) N (b) R (c) Z (d) C – {0} • In congruence modulo 5, {x  z / x = 5k + 2, k  z} represents(a) [0] (b) [5] (c) [7] (d) [2]

Choose The Correct Answer • [5] .12 [11] is(a) [55] (b) [12] (c) [7] (d) [11] • [3] +8 [7] is(a) [10] (b) [8] (c) [5] (d) [2]

Choose The Correct Answer • In the group {G, }, G = {1, – 1, i, – i }, order of – 1 is(a) – 1 (b) 1 (c) 2 (d) 0 • In the group {G, }, G = {1, – 1, i, – i }, order of – i is(a) 2 (b) 0 (c) 4 (d) 3

Choose The Correct Answer • In the group {G, }, G = {1, , 2},  is cube root of unity, order of 2 is(a) 2 (b) 1 (c) 4 (d) 3 • In the group {Z+, +4}, order of ([0]) is(a) 1 (b) (c) cannot be determined (d) 3

Choose The Correct Answer • In the group {Z+, +4}, O([3]) is(a) 4 (b) 3 (c) 2 (d) 1 • In (S, o), xoy = x, x, y  S, then ‘o’ is(a) only associative (b) only commutative(c) associative and commutative (d) neither associative nor commutative

Choose The Correct Answer • In (N, *), x * y = max{x, y} x, y  N then (N, *) is(a) only closed (b) only semi group (c) only monoid (d) a group • The set of positive even numbers, with usual multiplication forms(a) a finite group (b) only a semi group (c) only a monoid (d) an infinite group

Choose The Correct Answer • The set of positive even numbers, with usual addition forms(a) a finite group (b) only a semi group (c) only a monoid (d) an infinite group • In {Z5 – {[0]}, 5), O([3]) is(a) 5 (b) 3 (c) 4 (d) 2

Choose The Correct Answer • Which of the following statement is false? (G is a group)(a) For any element a  G, O(a) = O(a – 1)(b) Let a  G and O(a) = n, then am = e  m divides n, where m  Z(c) If O(a) = 2 for each a  G, then G is Abelian (a  e)(d) If a = a – 1 (or a2 = e) for each a G, then G is Abelian

Choose The Correct Answer • Which of the following is not true?(a) If (G, *) is Abelian then O(G) = 4 (b) If G is of even order then there is an element ‘a’ different from e such that a2 = e (equivalently a = a – 1 )(c) If O(G)  4, then G is Abelian(d) (a * b)2 = a2 * b2 for all a, b G  G is Abelian

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Discrete Mathematics