Discrete Mathematics Multiple Choice Questions 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
Answer: b
Clarification: 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
Answer: b
Clarification: 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
Answer: b
Clarification: Here, B1 is the group and identity element is 0, means for all a∈B1, a+n.0=a. As a2 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
Answer: c
Clarification: 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
Answer: a
Clarification: 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
Answer: b
Clarification: 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
Answer: a
Clarification: 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
Answer: b
Clarification: 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
Answer: c
Clarification: 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
Answer: a
Clarification: The set of even natural numbers is closed by the addition as the sum of any two of them produces another even number. Hence, this closed set satisfies the closure property.