Mathematics Questions and Answers – Binary Operations

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Binary Operations”.

1. Let a binary operation ‘*’ be defined on a set A. The operation will be commutative if ________
a) a*b=b*a
b) (a*b)*c=a*(b*c)
c) (b ο c)*a=(b*a) ο (c*a)
d) a*b=a
View Answer

Answer: a
Explanation: A binary operation ‘*’ defined on a set A is said to be commutative only if a*b=b*a, ∀a, b∈A.
If (a*b)*c=a*(b*c), then the operation is said to associative ∀ a, b∈ A.
If (b ο c)*a=(b*a) ο (c*a), then the operation is said to be distributive ∀ a, b, c ∈ A.
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2. Let a*b=6a4-9b4 be a binary operation on R, then * is commutative.
a) True
b) False
View Answer

Answer: b
Explanation: The given statement is false. The binary operation ‘*’ is commutative if a*b=b*a
Here, a*b=6a4-9b4 and b*a=6b4-9a4
⇒a*b≠b*a
Hence, the ‘*’ is not commutative.

3. Let ‘*’ be a binary operation on N defined by a*b=a-b+ab2, then find 4*5.
a) 9
b) 88
c) 98
d) 99
View Answer

Answer: d
Explanation: The binary operation is defined by a*b=a-b+ab2.
∴4*5=4-5+4(52)=-1+100=99.
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4. Let ‘*’ be defined on the set N. Which of the following are both commutative and associative?
a) a*b=a+b
b) a*b=a-b
c) a*b=ab2
d) a*b=ab
View Answer

Answer: a
Explanation: The binary operation ‘*’ is both commutative and associative for a*b=a+b.
The operation is commutative on a*b=a+b because a+b=b+a.
The operation is associative on a*b=a+b because (a+b)+c=a+(b+c).

5. Let ‘&’ be a binary operation defined on the set N. Which of the following definitions is commutative but not associative?
a) a & b=a-b
b) a & b=a+b
c) a & b=ab – 8
d) a & b=ab
View Answer

Answer: c
Explanation: The binary operation ‘&’ is commutative but not associative for a*b=ab-8.
For Commutative: a & b=ab-8 and b & a=ba-8
ab-8=ba-8. Hence, a & b=ab-8 is commutative.
For Associative: (a &b)& c=(ab-8)& c=(ab-8)c-8=abc-8c-8=abc-8c-8.
a& (b &c)=a&(bc-8)=a(bc-8)-8=abc-8a-8.
⇒(a&b) & c≠a& (b& c). Hence, the function is not associative.
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6. Let ‘*’ be a binary operation defined by a*b=4ab. Find (a*b)*a.
a) 4a2 b
b) 16a2 b
c) 16ab2
d) 4ab2
View Answer

Answer: b
Explanation: Given that, a*b=4ab.
Then, (a*b)*a=(4ab)*a
=4(4ab)(a)=16a2 b.

7. Let ‘*’ and ‘^’ be two binary operations such that a*b=a2 b and a ^ b = 2a+b. Find (2*3) ^ (6*7).
a) 256
b) 286
c) 276
d) 275
View Answer

Answer: c
Explanation: Given that, a*b=a2 b and a ^ b = 2a+b.
∴(2*3)^(6*7)=(22×3)^(62×7)
=12^252=2(12)+252=276.
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8. An element is said to be invertible only if there is an identity element in that binary operation.
a) True
b) False
View Answer

Answer: a
Explanation: The given statement is true. If there is a binary operation *:M×M → M with an identity element a∈ M is said to be invertible with respect to the binary operation * if there exists an element b ∈ M such that a*b = e = b*a, b is called inverse of a.

9. Let ‘*’ be a binary operation defined by a*b=3ab+5. Find 8*3.
a) 1547
b) 1458
c) 1448
d) 1541
View Answer

Answer: d
Explanation: It is given that a*b=3ab+5.
Then, 8*3=3(83)+5=3(512)+5=1536+5=1541.
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10. Which of the following is not a type of binary operation?
a) Transitive
b) Commutative
c) Associative
d) Distributive
View Answer

Answer: a
Explanation: Transitive is not a type of binary operation. It is a type of relation. Distributive, associative, commutative are different types of binary operations.

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

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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 | Youtube | Instagram | Facebook | Twitter