A data verification algorithm analyzes inputs based on their fundamental mathematical clas… — State PSC Exam

A data verification algorithm analyzes inputs based on their fundamental mathematical classifications. During a specific test run, the system is fed four distinct variables:

- $w = \frac{22}{7}$
- $x =$ an arbitrary irrational number
- $y =$ the additive identity of the real number system
- $z =$ an arbitrary prime number strictly greater than 2

Based on the strict mathematical definitions and properties of numbers, which of the following statements are ALWAYS true? (Select all that apply)

The product $x \cdot w$ is guaranteed to evaluate to an irrational number.Answer
The sum $z + y$ will always result in a positive odd integer.Answer
C
The quotient $\frac{\pi}{w}$ evaluates to exactly $1$ , as both terms represent the same mathematical constant.
D
The variable $y$ belongs to the set of positive integers.

Answer

The statements confirming that the product of

x

and

w

is irrational, and that the sum of

z

and

y

is a positive odd integer, are the mathematically correct conclusions.

The correct statements accurately apply the closure rule that a non-zero rational multiplied by an irrational is irrational, and correctly identify that any prime greater than 2 is a positive odd integer whose value is unchanged when adding zero.

Step-by-Step Solution

Classify the variable

w

and evaluate its product with

x

The number

w = \frac{22}{7}

is a non-zero rational number. The product of a non-zero rational number and an irrational number (

x

) is always irrational.

This tests the closure properties of rational and irrational numbers under multiplication.

Identify the value of

y

and evaluate its classification.

The additive identity of the real number system is

0

. The number

0

is an even integer, but it is neither positive nor negative.

This ensures the exact boundary conditions of integer classifications (positive, negative, zero) are understood.

Determine the properties of

z

and evaluate the sum

z + y

All prime numbers strictly greater than

2

are odd and positive. Since

y = 0

, the sum

z + 0 = z

remains a positive odd integer.

This applies the parity rules of prime numbers.

Compare

\pi

and

w

to evaluate their quotient.

While

\frac{22}{7}

is a common approximation for

\pi

\pi

is irrational and

\frac{22}{7}

is rational. They are not equal, so their quotient is not

1

This distinguishes between exact mathematical constants and their real-world practical approximations.

Key Concept

Classification of Real Numbers, Closure Properties, Parity of Primes, and Properties of Zero

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