Discrete Mathematics Questions and Answers – Groups – Closure and Associativity

«
»

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
View Answer

Answer: b
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.
advertisement

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

Answer: b
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
View Answer

Answer: b
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
View Answer

Answer: c
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
View Answer

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

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
View Answer

Answer: b
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
View Answer

Answer: a
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
View Answer

Answer: b
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
View Answer

Answer: c
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….}.
advertisement

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
View Answer

Answer: a
Explanation: 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.

Sanfoundry Global Education & Learning Series – Discrete Mathematics.

To practice all areas of Discrete Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers.

advertisement
advertisement
advertisement
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn