Discrete Mathematics Questions and Answers – Groups – Existence of Identity & Inverse

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This set of Discrete Mathematics Puzzles focuses on “Groups – Existence of Identity & Inverse”.

1. In a group there must be only __________ element.
a) 1
b) 2
c) 3
d) 5
View Answer

Answer: a
Explanation: There can be only one identity element in a group and each element in a group has exactly one inverse element. Hence, two important consequences of the group axioms are the uniqueness of the identity element and the uniqueness of inverse elements.
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2. _____ is the multiplicative identity of natural numbers.
a) 0
b) -1
c) 1
d) 2
View Answer

Answer: c
Explanation: 1 is the multiplicative identity of natural numbers as a⋅1=a=1⋅a ∀a∈N. Thus, 1 is the identity of multiplication for the set of integers(Z), set of rational numbers(Q), and set of real numbers(R).

3. An identity element of a group has ______ element.
a) associative
b) commutative
c) inverse
d) homomorphic
View Answer

Answer: c
Explanation: By the definition of all elements of a group have an inverse. For an element, a in a group G, an inverse of a is an element b such that ab=e, where e is the identity in the group. The inverse of an element is unique and usually denoted as -a.

4. __________ matrices do not have multiplicative inverses.
a) non-singular
b) singular
c) triangular
d) inverse
View Answer

Answer: b
Explanation: The rational numbers are an extension of the integer numbers in which each non-zero number has an inverse under multiplication. A 3 × 3 matrix may or may not have an inverse under matrix multiplication. The matrices which do not have multiplicative inverses are termed as singular matrices.

5. If X is an idempotent nonsingular matrix, then X must be ___________
a) singular matrix
b) identity matrix
c) idempotent matrix
d) nonsingular matrix
View Answer

Answer: b
Explanation: Since X is idempotent, we have X2=X. As X is nonsingular, it is invertible. Thus, the inverse matrix X-1 exists. Then we have, I=X-1X = X-1X2=IX=X.
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6. If A, B, and C are invertible matrices, the expression (AB-1)-1(CA-1)-1C2 evaluates to ____________
a) BC
b) C-1BC
c) AB-1
d) C-1B
View Answer

Answer: a
Explanation: Using the properties (AB)-1=b-1A-1 and (A-1)-1=A, we may have,
(AB-1)-1(CA-1)-1C2
=(B-1)-1A-1(A-1)-1C-1C2
=BA-1AC-1C2
=BIC=BC [As, A-1A=I].

7. If the sum of elements in each row of an n×n matrix Z is zero, then the matrix is ______________
a) inverse
b) non-singular
c) additive inverse
d) singular
View Answer

Answer: d
Explanation: By the definition, an n×n matrix A is said to be singular if there exists a nonzero vector v such that Av=0. Otherwise, it is known that A is a nonsingular matrix.

8. ___________ are the symmetry groups used in the Standard model.
a) lie groups
b) subgroups
c) cyclic groups
d) poincare groups
View Answer

Answer: a
Explanation: A symmetry group can encode symmetry features of a geometrical object. The group consists of the set of transformations that leave the object unchanged. Lie groups are such symmetry groups used in the standard model of particle physics.

9. A semigroup S under binary operation * that has an identity is called __________
a) multiplicative identity
b) monoid
c) subgroup
d) homomorphism
View Answer

Answer: b
Explanation: Let P(S) is a commutative semigroup has the identity e, since e*A=A=A*e for any element A belongs to P(S). Hence, P(S) is a monoid.
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10. An element a in a monoid is called an idempotent if ______________
a) a-1=a*a-1
b) a*a2=a
c) a2=a*a=a
d) a3=a*a
View Answer

Answer: c
Explanation: An algebraic structure with a single associative binary operation and an Identity element are termed as a monoid. It is studied in semigroup theory. An element x in a monoid is called idempotent if a2 = a*a = a.

Sanfoundry Global Education & Learning Series – Discrete Mathematics.

To practice all Puzzles on 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