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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.

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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

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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.

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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.

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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.

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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.

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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).

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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).

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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.

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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.}

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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

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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.

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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.

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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|.

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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} *).

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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.

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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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