MATH-105 Contemporary Mathematics FINAL READINESS EXAM Question & Answers

MATH-105 Contemporary Mathematics FINAL READINESS EXAM Question & Answers
1. : In the context of group theory, which of the
following sets along with the operation given forms a group?
 - A. The set of all integers with addition.
 - B. The set of all positive integers with multiplication.
 - C. The set of non-zero rational numbers with
multiplication.
 - D. The set of all real numbers excluding zero with
multiplication.
 Answer: C. The set of non-zero rational numbers with
multiplication.
 Rationale: This set forms a group because it satisfies
all group properties: closure, associativity, identity (1), and
inverses (e.g., for any rational number `a/b`, its inverse is
`b/a`).
2. Question: Which of the following sequences converges
to zero as \( n \to \infty \)?
 - A. \( a_n = \frac{1}{n} \)
 - B. \( b_n = \frac{n}{n+1} \)
 - C. \( c_n = (-1)^n \)
 - D. \( d_n = \sin(n) \)
 Answer: A. \( a_n = \frac{1}{n} \)
 Rationale: As \( n \to \infty \), \(\frac{1}{n} \to 0\). The
other sequences either oscillate or converge to values other
than zero.
3. Question: Which type of infinity is larger in the context
of set theory?
 - A. The cardinality of the set of natural numbers.
 - B. The cardinality of the set of real numbers.
 Answer: B. The cardinality of the set of real numbers.
 Rationale: The real numbers have a higher cardinality
(continuum) compared to the set of natural numbers, which
is countably infinite.