Discrete Mathematics | Group MCQs

Discrete Mathematics | Group MCQs: This section contains multiple-choice questions and answers on Group in Discrete Mathematics.
Submitted by Anushree Goswami, on October 30, 2022

1. Consider G a non-void set where every pair of ordered elements of G will have an element of G denoted by a * b. How many properties are true, when we can say that G is a binary group?

  1. One
  2. Two
  3. Three
  4. Four

Answer: C) Three

Explanation:

Consider G a non-void set where every pair of ordered elements of G will have an element of G denoted by a * b. The three properties are needed to be true, then we can say that G is a binary group.


2. Consider G a non-void set where every pair of ordered elements of G will have an element of G denoted by a * b. If the following property/ies are true, then we can say that G is a binary group -

  1. Associativity
  2. Identity
  3. Inverse
  4. All of the above

Answer: D) All of the above

Explanation:

Consider G a non-void set where every pair of ordered elements of G will have an element of G denoted by a * b. If the following property/ies are true, then we can say that G is a binary group -

  1. Associativity
  2. Identity
  3. Inverse

3. Associative property on binary operation * states that -

  1. a*(b*c)=(a*b)*c, ∀ a,b,c ∈ G
  2. a*e=e*a=a, ∀ a ∈ G
  3. a*b=b*a=e, ∀ a, b ∈ G
  4. None

Answer: A) a*(b*c)=(a*b)*c, ∀ a,b,c ∈ G

Explanation:

Associative property on binary operation * states that a*(b*c)=(a*b)*c, ∀ a,b,c ∈ G.


4. Identity property on binary operation * states that -

  1. a*(b*c)=(a*b)*c, ∀ a,b,c ∈ G
  2. a*e=e*a=a, ∀ a ∈ G
  3. a*b=b*a=e, ∀ a, b ∈ G
  4. None

Answer: B) a*e=e*a=a, ∀ a ∈ G

Explanation:

Identity property on binary operation * states that a*e=e*a=a, ∀ a ∈ G.


5. Inverse property on binary operation * states that -

  1. a*(b*c)=(a*b)*c, ∀ a,b,c ∈ G
  2. a*e=e*a=a, ∀ a ∈ G
  3. a*b=b*a=e, ∀ a, b ∈ G
  4. None

Answer: C) a*b=b*a=e, ∀ a, b ∈ G

Explanation:

Inverse property on binary operation * states that a*b=b*a=e, ∀ a, b ∈ G.


6. The group is called abelian if it has the property of ____law.

  1. Associative
  2. Identity
  3. Inverse
  4. Commutative

Answer: D) Commutative

Explanation:

The group is called abelian if it has the property of commutative law.


7. ___ identity element exists in a Group G (unique identity).

  1. One
  2. Two
  3. Three
  4. Four

Answer: A) One

Explanation:

One identity element exists in a Group G (unique identity).


8. If a is unique in a group G, then b is unique in G so that ____ (uniqueness if inverses).

  1. ab = ba
  2. ab = e
  3. ba = e
  4. ab = ba = e

Answer: D) ab = ba = e

Explanation:

If a is unique in a group G, then b is unique in G so that ab = ba = e (uniqueness if inverses).


9. The Group G is defined as ____,∀ a∈ G.

  1. (a-1)=a
  2. (a)-1=a
  3. (a-1)-1=a
  4. (a-1-1)-1=a

Answer: C) (a-1)-1=a

Explanation:

The Group G is defined as (a-1)-1=a,∀ a∈ G.


10. As part of Group G, ____,∀ a,b∈ G.

  1. (a b)=b-1a-1
  2. (a b-1)=ba-1
  3. (a b-1)=b-1a
  4. (a b-1)=b-1a-1

Answer: C) (a b-1)=b-1a-1

Explanation:

As part of Group G, (a b-1)=b-1a-1,∀ a,b∈ G.


11. It follows that the left and right cancellation laws apply to group G, which means -

  1. ab = ac implies b=c
  2. ba=ca implies b=c
  3. Both A and B
  4. None of the above

Answer: C) Both A and B

Explanation:

It follows that the left and right cancellation laws apply to group G, which means -

  1. ab = ac implies b=c
  2. ba=ca implies b=c

12. ____ groups are those for which the set G is ____.

  1. Finite, finite
  2. Infinite, infinite
  3. Both A and B
  4. None of the above

Answer: C) Both A and B

Explanation:

  1. Finite groups are those for which the set G is finite.
  2. Infinite groups are those for which the set G is infinite.

13. Counting the elements in the group G determines the ____ of the group.

  1. Number
  2. Elements
  3. Order
  4. Pair

Answer: C) Order

Explanation:

Counting the elements in the group G determines the order of the group.


14. Order of group G is denoted by -

  1. <G>
  2. |G|
  3. G
  4. *G

Answer: C) |G|

Explanation:

Order of group G is denoted by |G|.


15. There is only one identity element in an order _ group, i.e., ({e} *).

  1. 1
  2. 2
  3. 3
  4. 4

Answer: A) 1

Explanation:

There is only one identity element in an order 1 group, i.e., ({e} *).


16. There are two elements in a group of order 2, namely, ____.

  1. Identity, Other
  2. Identity, Identity
  3. Inverse, Inverse
  4. Associative, Inverse

Answer: A) Identity, Other

Explanation:

There are two elements in a group of order 2, namely, one identity element and one other element.


17. Three elements make up order 3, namely an ____ element and two ___ elements.

  1. Identity, Identity
  2. Inverse, Associative
  3. Associative, Identity
  4. Identity, Other

Answer: D) Identity, Other

Explanation:

Three elements make up order 3, namely an identity element and two other elements.





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