Number Classification and Set Theory

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Questions and Answers

Which of the following numbers is classified as a natural number?

2 (correct)

1.333333...

-7

-0.5

What classification does the number -16 belong to?

Rational number

Natural number

Integer (correct)

Irrational number

Identify the set represented by *x: x is an integer smaller than 5+.

-5, -4, -3, -2, -1, 0, 1, 2, 3, 4

-2, -1, 0, 1, 2, 3, 4 (correct)

0, 1, 2, 3, 4, 5

1, 2, 3, 4

Which option correctly lists the elements of the set *x: x is a positive integer multiple of three+?

3, 6, 9, 12, ... Signup and view all the answers

What is the set represented by *x: x = y² where y is an integer+?

0, 1, 4, 9, 16, ... Signup and view all the answers

Which expression represents the solution set for *x: (3x - 1)(x + 2) = 0+?

-2, 1 Signup and view all the answers

Determine the elements of the set *x: 2x is a positive integer+.

2, 4, 6, 8, ... Signup and view all the answers

Which of the following is a rational number?

-0.5 Signup and view all the answers

What does the set {z: z ∈ X or -z ∈ X} equal if X = {0, 1, 2}?

{0, 1, -1, 2, -2} Signup and view all the answers

What is the cardinality of the set {x: x is an integer and 1/8 < x < 17/2}?

8 Signup and view all the answers

Which option correctly represents the set {3, 6, 9, 12, 15, ..., 30}?

{x: x = 3z and 1 ≤ z ≤ 10} Signup and view all the answers

How can the set {2, 3, 5, 7, 11, 13, 17, 19, 23, ...} be characterized?

{x: x is a prime number} Signup and view all the answers

Which one of the following statements is actually true?

2 ∈ {1, 2, 3, 4, 5} Signup and view all the answers

Given ℧ and sets A and B, which of the following is correct?

Neither Signup and view all the answers

How many students study none of the three subjects if 18 students are surveyed?

6 Signup and view all the answers

What does the set A ∩ B equal if A = {x ∈ ℧: x is less than 7} and B = {x ∈ ℧: x is a multiple of 3}?

{3, 6} Signup and view all the answers

Which statement about the set operations is incorrect?

A + B includes elements found in A or B. Signup and view all the answers

Which of the following subsets can be derived from the set {1,2,3}?

{1, 3} Signup and view all the answers

Study Notes

Number Classification

Natural numbers (ℕ) are positive integers starting from 1: {1, 2, 3, 4,...}
Integers (ℤ) include positive and negative whole numbers and zero: {... , -2, -1, 0, 1, 2,...}
Rational numbers (ℚ) are numbers that can be expressed as a fraction where numerator and denominator are integers: {𝑎/𝑏: 𝑎, 𝑏 ∈ ℤ, 𝑏 ≠ 0}
Irrational numbers (ℚ𝑐) are real numbers that are not rational.
Real numbers (ℝ) consist of all rational and irrational numbers.

Set Notation and Representations

{𝑥: 𝑥 is an integer and −3 < 𝑥 < 4} = {-2, -1, 0, 1, 2, 3}
{𝑥: 𝑥 is a positive integer multiple of three} = {3, 6, 9, 12, ...}
{𝑥: 𝑥 is an integer smaller than 5} = {... , -2, -1, 0, 1, 2, 3, 4}
{𝑥: 𝑥 is a rational number with denominator 2} = {... , -3/2, -1/2, 0/2, 1/2, 3/2, ...}

Set Operations

The Set *𝑥: 𝑥 = 𝑦^2 and 𝑦 is an integer = {0, 1, 4, 9, 16, ...}
The set *𝑥: 𝑥 is an integer and (3𝑥 − 1)(𝑥 + 2) = 0 = {-2, 1/3}
The set *𝑥: 2𝑥 is a positive integer = {1, 2, 3, 4, ...}
If 𝑋 = {0, 1, 2}, then *𝑧: 𝑧 ∈ 𝑋 or −𝑧 ∈ 𝑋 = {0, 1, -1, 2, -2}
The set *𝑥: 𝑥 is an integer and 1/8 < 𝑥 < 17/2 has the cardinality 8
The set {3,6,9,12,15,...,27,30} = {𝑥: 𝑥 = 3𝑧 and 1 ≤ 𝑧 ≤ 10}
The set {2,3,5,7,11,13,17,19,23,...} = {𝑥: 𝑥 is a prime number}

Set Relations

2 ∈ {1,2,3,4,5} - True
{2} ∈ {1,2,3,4,5} - False
2 ⊆ {1, 2, 3, 4, 5} - True
{2} ⊆ {1, 2, 3, 4, 5} - True
∅ ⊆ {∅, {∅}} - True
{∅} ⊆ {∅, {∅}} - True
0 ∈ ∅ - False
{1,2,3,4,5} = {5,4,3,2,1} - True

Subsets

Subsets of {𝑎} = {∅, {𝑎}}
Subsets of {𝑎, 𝑏} = {∅, {𝑎}, {𝑏}, {𝑎, 𝑏}}
Subsets of {𝑎, 𝑏, 𝑐} = {∅, {𝑎}, {𝑏}, {𝑐}, {𝑎, 𝑏}, {𝑎, 𝑐}, {𝑏, 𝑐}, {𝑎, 𝑏, 𝑐}}
A set with 𝑛 elements will have 2^𝑛 subsets
The empty set has one subset, which is itself

Set Membership & Subset Relations

𝑥 = {1}; 𝐴 = {1, 2, 3} - 𝑥 ∈ 𝐴
𝑥 = {1}; 𝐴 = {{1}, {2}, {3}} - 𝑥 ⊆ 𝐴
𝑥 = {1}; 𝐴 = {1, 2, {1, 2}} - 𝑥 ∈ 𝐴 and 𝑥 ⊆ 𝐴
𝑥 = {1, 2}; 𝐴 = {1, 2, {1, 2}} - 𝑥 ∈ 𝐴 and 𝑥 ⊆ 𝐴
𝑥 = {1}; 𝐴 = {{1, 2, 3}} - Neither 𝑥 ∈ 𝐴 nor 𝑥 ⊆ 𝐴
𝑥 = 1; 𝐴 = {{1}, {2}, {3}} - Neither 𝑥 ∈ 𝐴 nor 𝑥 ⊆ 𝐴

Set Operations - Union, Intersection, Complement, Difference

Let ℧ = {1,2,3,4,5,6,7,8,9,10}, 𝐴 = {𝑥 ∈ ℧: 𝑥 is less than 7}, 𝐵 = {𝑥 ∈ ℧: 𝑥 is a multiple of 3}
𝐴 ∪ 𝐵 = {1,2,3,4,5,6,9}
𝐴 ∩ 𝐵 = {3, 6}
𝐴𝑐 = {7, 8, 9, 10}
𝐵𝑐 = {1, 2, 4, 5, 7, 8, 10}
𝐴\B = {1, 2, 4, 5}

Set Relationships

Let ℧ = {𝑥: 𝑥 is an integer and 2 ≤ 𝑥 ≤ 10}, 𝐴 = {𝑥: 𝑥 is odd}, 𝐵 = {𝑥: 𝑥 is a multiple of 3}
𝐴 ⊂ 𝐵 is not true
𝐵 ⊂ 𝐴 is not true
𝐴 = 𝐵 is not true
Neither 𝐴 ⊂ 𝐵, 𝐵 ⊂ 𝐴 nor 𝐴 = 𝐵 is true
Let ℧ = {𝑥: 𝑥 is an integer and 2 ≤ 𝑥 ≤ 10}, 𝐴 = {𝑥: 𝑥 ∈ ℤ}, 𝐵 = {𝑥: 𝑥 is a power of 2 or 3}
𝐴 ⊂ 𝐵 is not true
𝐵 ⊂ 𝐴 is not true
𝐴 = 𝐵 is not true
Neither 𝐴 ⊂ 𝐵, 𝐵 ⊂ 𝐴 nor 𝐴 = 𝐵 is true

Student Survey Problem

There are 18 students in a room
7 study mathematics, 10 study science, and 10 study computer programming.
3 study mathematics and science, 4 study mathematics and computer programming, and 5 study science and computer programming.
1 student studies all three subjects.
It can be concluded that 4 students study none of the three subjects.

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Description

This quiz explores the concepts of number classification, including natural, integer, rational, irrational, and real numbers. It also covers set notation and representations, as well as set operations, equipping you with a solid understanding of these mathematical foundations.