Discrete Mathematics Questions and Answers – Group Axioms

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This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Group Axioms”.

1. __________ are called group postulates.
a) Group lemmas
b) Group theories
c) Group axioms
d) Group
View Answer

Answer: c
Explanation: The group axioms are also called the group postulates. A group with an identity (that is, a monoid) in which every element has an inverse is termed as semi group.
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2. A subgroup has the properties of ________
a) Closure, associative
b) Commutative, associative, closure
c) Inverse, identity, associative
d) Closure, associative, Identity, Inverse
View Answer

Answer: d
Explanation: A subgroup S is a subset of a group G (denoted by S <= G) if it holds the four properties simultaneously – Closure, Associative, Identity and Inverse element.

3. If a * b = a such that a ∗ (b ∗ c) = a ∗ b = a and (a * b) * c = a * b = a then ________
a) * is associative
b) * is commutative
c) * is closure
d) * is abelian
View Answer

Answer: a
Explanation: ‘∗’ can be defined by the formula a∗b = a for any a and b in S. Hence, (a ∗ b)∗c = a∗c = a and a ∗(b ∗ c)= a ∗ b = a. Therefore, ”∗” is associative. Hence (S, ∗) is a semigroup. On the contrary, * is not associative since, for example, (b•c)•c = a•c = c but b•(c•c) = b•a = b Thus (S,•) is not a semigroup.

4. The set of odd and even positive integers closed under multiplication is ________
a) a free semigroup of (M, ×)
b) a subsemigroup of (M, ×)
c) a semigroup of (M, ×)
d) a subgroup of (M, ×)
View Answer

Answer: b
Explanation: Let C and D be the set of even and odd positive integers. Then, (C, ×) and (D, ×) are subsemigroups of (M, ×) since A and B are closed under multiplication. On the other hand, (A, +) is a subsemigroup of (N, +) since A is closed under addition, but (B, +) is not a subsemigroup of (N, +) since B is not closed under addition.

5. If F is a free semigroup on a set S, then the concatenation of two even words is ________
a) a semigroup of F
b) a subgroup of F
c) monoid of F
d) cyclic group of F
View Answer

Answer: b
Explanation: Let F be the free semigroup on the set S = {m,n}. Let, E consist of all even words, i.e, words with even length and the concatenation of two such words is also even. Thus E is a subsemigroup of F.
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6. The set of rational numbers form an abelian group under _________
a) Association
b) Closure
c) Multiplication
d) Addition
View Answer

Answer: c
Explanation: The set of nonzero rational numbers form an abelian group under multiplication. The number 1 is the identity element and q/p is the multiplicative inverse of the rational number p/q.

7. Condition of semigroup homomorphism should be ____________
a) f(x * x) = f(x * y)
b) f(x) = f(y)
c) f(x) * f(y) = f(y)
d) f(x * y) = f(x) * f(y)
View Answer

Answer: d
Explanation: Consider two semigroups (S,∗) and (S’,∗’). A function f: S -> S’ is called a semigroup homomorphism if f(a∗b) = f(a)∗f(b). Suppose f is also one-to-one and onto. Then f is called an isomorphism between S and S’ and S and S’ are said to be isomorphic semigroups.

8. A function f:(M,∗)→(N,×) is a homomorphism if ______
a) f(a, b) = a*b
b) f(a, b) = a/b
c) f(a, b) = f(a)+f(b)
d) f(a, b) = f(a)*f(a)
View Answer

Answer: b
Explanation: The function f is a homomorphism since f(x∗y)= f(ac, bd)= (ac)/(bd) = (a/b)(c/d) = f(x)f(y).

9. A function defined by f(x)=2*x such that f(x+y)=2x+y under the group of real numbers, then ________
a) Isomorphism exists
b) Homomorphism exists
c) Heteromorphic exists
d) Association exists
View Answer

Answer: b
Explanation: Let T be the group of real numbers under addition, and let T’ be the group of positive real numbers under multiplication. The mapping f: T -> T’ defined by f(a)=2*a is a homomorphism because f(a+b)=2a+b = 2a*2b = f(a)*f(b). Again f is also one-to-one and onto T and T’ are isomorphic.
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10. If x * y = x + y + xy then (G, *) is _____________
a) Monoid
b) Abelian group
c) Commutative semigroup
d) Cyclic group
View Answer

Answer: c
Explanation: Let x and y belongs to a group G.Here closure and associativity axiom holds simultaneously. Let e be an element in G such that x * e = x then x+e+xe=a => e(1+x)=0 => e = 0/(1+x) = 0. So, identity axiom does not exist but commutative property holds. Thus, (G,*) is a commutative semigroup.

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.

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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