# Discrete Mathematics Questions and Answers – Groups – Closure and Associativity

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This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Groups – Closure and Associativity”.

1. Let (A7, ⊗7)=({1, 2, 3, 4, 5, 6}, ⊗7) is a group. It has two sub groups X and Y. X={1, 3, 6}, Y={2, 3, 5}. What is the order of union of subgroups?
a) 65
b) 5
c) 32
d) 18

Explanation: Given, (A7, ⊗7)=({1, 2, 3, 4, 5, 6}, ⊗7) and the union of two sub groups X and Y, X={1, 3, 6} Y={2, 3, 5} is X∪Y={1, 2, 3, 5, 6} i.e., 5. Here, the order of the union can not be divided by order of the group.

2. A relation (34 × 78) × 57 = 57 × (78 × 34) can have __________ property.
a) distributive
b) associative
c) commutative
d) closure

Explanation: For any three elements(numbers) a, b and c associative property describes a × ( b × c ) = ( a × b ) × c [for multiplication]. Hence associative property is true for multiplication and it is true for multiplication also.

3. B1: ({0, 1, 2….(n-1)}, xm) where xn stands for “multiplication-modulo-n” and B2: ({0, 1, 2….n}, xn) where xn stands for “multiplication-modulo-m” are the two statements. Both B1 and B2 are considered to be __________
a) groups
b) semigroups
c) subgroups
d) associative subgroup

Explanation: Here, B1 is the group and identity element is 0, means for all a∈B1, a+n.0=a. As a<n. But in B2 identity element does not exist. Here, 0 can not be the identity element. For example, for one of the member n of the set we have n+n.0=0, It will be n. So, B2 is not a group. Both B1 and B2 are semigroups as they satisfy closure and associativity property.

4. If group G has 65 elements and it has two subgroups namely K and L with order 14 and 30. What can be order of K intersection L?
a) 10
b) 42
c) 5
d) 35

Explanation: As it is an intersection so the order must divide both K and L. Here 3, 6, 30 does not divide 14. But 5 must be the order of the group as it divides the order of intersection of K and L as well as the order of the group.

5. Consider the binary operations on X, a*b = a+b+4, for a, b ∈ X. It satisfies the properties of _______
a) abelian group
b) semigroup
c) multiplicative group
d) isomorphic group

Explanation: Since * closed operation, a*b belongs to X. Hence, it is an abelian group.

6. Let * be the binary operation on the rational number given by a*b=a+b+ab. Which of the following property does not exist for the group?
a) closure property
b) identity property
c) symmetric property
d) associative property

Explanation: For identity e, a+e=e+a=e, a*e = a+e+ae = a => e=0 and e+a = e+a+ea = a => e=0. So e=0 will be identity, for e to be identity, a*e = a ⇒ a+e+ae = a ⇒ e+ae = 0 and e(1+a) = 0 which gives e=0 or a=-1. So, when a = -1, no identity element exist as e can be any value in that case.

7. Let G be a finite group with two sub groups M & N such that |M|=56 and |N|=123. Determine the value of |M⋂N|.
a) 1
b) 56
c) 14
d) 78

Explanation: We know that gcd(56, 123)=1. So, the value of |M⋂N|=1.

8. A group G, ({0}, +) under addition operation satisfies which of the following properties?
a) identity, multiplicity and inverse
b) closure, associativity, inverse and identity
c) multiplicity, associativity and closure
d) inverse and closure

Explanation: Closure for all a, b∈G, the result of the operation, a+b, is also in G. Since there is one element, hence a=b=0, and a+b=0+0=0∈G. Hence, closure property is satisfied. Associative for all a, b, c∈G, (a+b)+c=a+(b+c). For example, a=b=c=0. Hence (a+b)+c=a+(b+c)
⟹(0+0)+0=0+(0+0)⟹0=0. Hence, associativity property is satisfied. Suppose for an element e∈G such that, there exists an element a∈G and so the equation e+a=a+e=a holds. Such an element is unique, the identity element property is satisfied. For example, a=e=0. Hence e+a = a+e⟹0+0=0+0⟹0=a. Hence e=0 is the identity element. For each a∈G, there exists an element b∈G (denoted as a-1), such that a+b=b+a=e, where e is the identity element. The inverse element is 0 as the addition of 0 with 0 will be 0, which is also an identity element of the structure.

9. If (M, *) is a cyclic group of order 73, then number of generator of G is equal to ______
a) 89
b) 23
c) 72
d) 17

Explanation: We need to find the number of co-primes of 73 which are less than 73. As 73 itself is a prime, all the numbers less than that are co-prime to it and it makes a group of order 72 then it can be of {1, 3, 5, 7, 11….}.

10. The set of even natural numbers, {6, 8, 10, 12,..,} is closed under addition operation. Which of the following properties will it satisfy?
a) closure property
b) associative property
c) symmetric property
d) identity property 