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Discrete Mathematics | Binary Operations Properties MCQs

Discrete Mathematics | Binary Operations Properties MCQs: This section contains multiple-choice questions and answers on Binary Operations Properties in Discrete Mathematics.
Submitted by Anushree Goswami, on October 28, 2022

1. Which of the following is a/the property/ies of binary operations?

1. Closure Property
2. Associative Property
3. Commutative Property
4. All of the above

Answer: D) All of the above

Explanation:

The following are the properties of binary operations -

1. Closure Property
2. Associative Property
3. Commutative Property

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2. A non-empty set A and a binary operation * on A are closed under the operation *, if ___ ∈ A, where a and b are elements of A.

1. a*b
2. a+b
3. a-b
4. a/b

Answer: A) a*b

Explanation:

A non-empty set A and a binary operation * on A are closed under the operation *, if a * b ∈ A, where a and b are elements of A.

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3. There is a non-empty set A, then a binary operation * on A is associative, if for every a, b, c, ∈ A, we have ____.

1. (a + b) * c = a* (b*c)
2. (a * b) * c = a* (b*c)
3. (a / b) * c = a* (b*c)
4. (a = b) * c = a* (b*c)

Answer: B) (a * b) * c = a* (b*c)

Explanation:

There is a non-empty set A, then a binary operation * on A is associative, if for every a, b, c, ∈ A, we have (a * b) * c = a* (b*c).

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4. A non-empty set A gives rise to commutative binary operations, if for each a, b, ∈ A, we have ____.

1. a + b = b * a
2. a * b = b + a
3. a - b = b * a
4. a * b = b * a

Answer: D) a * b = b * a

Explanation:

A non-empty set A gives rise to commutative binary operations, if for each a, b, ∈ A, we have a * b = b * a.

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5. If we have a non-empty set A, then we have an identity property when e exists in A, and ____ = a ∀ a ∈ A.

1. a * e (left identity) = e * a (left identity)
2. a * e (right identity) = e * a (right identity)
3. a * e (right identity) = e * a (left identity)
4. e * e (right identity) = e * a (left identity)

Answer: C) a * e (right identity) = e * a (left identity)

Explanation:

If we have a non-empty set A, then we have an identity property when e exists in A, and a * e (right identity) = e * a (left identity) = a ∀ a ∈ A.

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6. The operation is the inverse property for a non-empty set A if ∃ an element b in A such that ____ = e, where b is called an inverse of a.

1. a * b (left inverse) = b * a (left inverse)
2. a * b (right inverse) = b * a (right inverse)
3. a * b (right inverse) = b * a (left inverse)
4. a * b (right inverse) = e * a (left inverse)

Answer: C) a * b (right inverse) = b * a (left inverse)

Explanation:

The operation is the inverse property for a non-empty set A if ∃ an element b in A such that a * b (right inverse) = b * a (left inverse) = e, where b is called an inverse of a.

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7. There is a non-empty set A, then the operation * has the ____ property, if for each a ∈A, we have a * a = a ∀ a ∈A.

1. Identity
2. Idempotent
3. Individual
4. Instinctive

Answer: B) Idempotent

Explanation:

There is a non-empty set A, then the operation * has the idempotent property, if for each a ∈A, we have a * a = a ∀ a ∈A.

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8. We are given a non-empty set A and we are given a binary operation * on A. Then the operation * distributes over +, assuming for each a, b, c ∈A, we have -

1. a * (b + c) = (a * b) + (a * c)
2. (b + c) * a = (b * a) + (c * a)
3. Both A and B
4. None of the above

Answer: C) Both A and B

Explanation:

We are given a non-empty set A and we are given a binary operation * on A. Then the operation * distributes over +, assuming for each a, b, c ∈A, we have -

1. a * (b + c) = (a * b) + (a * c)
2. (b + c) * a = (b * a) + (c * a)

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9. a * (b + c) = (a * b) + (a * c) is -

1. Distributivity
2. Left distributivity
3. Right distributivity
4. None of the above

Answer: B) Left distributivity

Explanation:

a * (b + c) = (a * b) + (a * c) is left distributivity.

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10. We are given a non-empty set A and we are given a binary operation * on A, then the operation * has the cancellation property, if for every a, b, c ∈A, we have -

1. a * b = a * c ⇒ b = c
2. b * a = c * a ⇒ b = c
3. Both A and B
4. None of the above

Answer: C) Both A and B

Explanation:

We are given a non-empty set A and we are given a binary operation * on A, then the operation * has the cancellation property, if for every a, b, c ∈A, we have -

1. a * b = a * c ⇒ b = c
2. b * a = c * a ⇒ b = c

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11. b * a = c * a ⇒ b = c is -

1. Cancellation
2. Left Cancellation
3. Right Cancellation
4. None of the above

Answer: C) Right Cancellation

Explanation:

b * a = c * a ⇒ b = c is right cancellation.

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