Discrete Mathematics: Sets and Elements

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Questions and Answers

What does the cardinality of a set measure?

The maximum element in the set
The average of all elements in the set
The sum of all elements in the set
The number of elements in the set (correct)

If A = {1,2,3,4,5} and B = {3,4,5,6,7}, what is A ∩ B?

{3,4}
{1,2}
{3,4,5} (correct)
{4,5}

If A = {x, y} and B = {x, y, z}, what is A ⊆ B?

A is not a subset of B
A is equivalent to B
A is the power set of B
A is a proper subset of B (correct)

What does it mean if a set has a cardinality of 0?

The set is the empty set (B) Signup and view all the answers

If A = {1,2,3} and B = {3,4,5}, what is A ∪ B?

{1,2,3,4} (B) Signup and view all the answers

Which set contains only positive numbers?

Natural numbers (N) (B) Signup and view all the answers

What is the cardinality of the set {, {a} , {b} }?

3 (B) Signup and view all the answers

Which set includes zero along with all natural numbers?

Whole numbers (W) (D) Signup and view all the answers

What is the correct roster form for the set of real numbers?

{47.3, -12, , …} (C) Signup and view all the answers

Which form of representation is used for defining standard sets like whole numbers?

All of the above (D) Signup and view all the answers

What is the set builder form for the natural numbers (N)?

{x :x is a counting number starting from 1} (A) Signup and view all the answers

Which set is denoted by the symbol Z or I and includes negative natural numbers, zero, and positive natural numbers?

Set Z or I (B) Signup and view all the answers

What is the set builder form for the set of even natural numbers?

{x: x is a natural number, which are divisible by 2} (D) Signup and view all the answers

Which set includes natural numbers that are not divisible by 2?

Set O (B) Signup and view all the answers

What is the roster form for the set of integers?

{..., -2, -1, 0, 1, 2, ...} (D) Signup and view all the answers

Which type of number can be expressed as a fraction with a non-zero denominator?

Rational number (B) Signup and view all the answers

What is the statement form for the set of odd natural numbers?

{1, 3, 5, 7, 9, ...} (B) Signup and view all the answers

Which of the following numbers is rational?

0.09009000900009... (B) Signup and view all the answers

What type of number is √2 / √2?

Rational (C) Signup and view all the answers

Is the number -12 rational or irrational?

Rational, because -12/1 can be expressed as a quotient of two integers. (C) Signup and view all the answers

Why is √3 considered irrational?

Because it cannot be expressed as a quotient of integers. (C) Signup and view all the answers

What category does the number √9 / 25 fall into?

Rational (C) Signup and view all the answers

Why is π/π considered rational?

Because both the numerator and denominator are the same value. (D) Signup and view all the answers

Is √2 a rational number?

No (B) Signup and view all the answers

Which of the following best describes a rational number?

A number that can be simplified to the quotient of two integers (D) Signup and view all the answers

What does the vertical bar '|' in set-builder notation represent?

Such that/where (D) Signup and view all the answers

Which symbol denotes 'is an element of' in set theory?

∈ (C) Signup and view all the answers

If a set contains all natural numbers greater than 7, how could it be represented in set-builder notation?

{x | x > 7} (C) Signup and view all the answers

What does '∧' represent in set theory?

Logical AND (B) Signup and view all the answers

Study Notes

Set Basics

A set can be either finite or infinite.
Example of a finite set: A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
Example of an infinite set: Z+ = {1, 2, 3, 4….} (set of positive integers)

Set Notation

∈ denotes "is an element of"
∉ denotes "is not an element of"
Example: Yellow ∈ C (Yellow is an element of set C)
Example: Violet ∉ C (Violet is not an element of set C)

Empty Set

∅ denotes an empty set (a set with no elements)
Example: A = ∅ (set A has no elements)

Cardinality

The cardinality of a set is a measure of the number of elements in the set.
Example: The cardinality of set A = {1, 2, 3} is 3.

Standard Sets of Numbers

Z or I: Set of integers (negative, zero, and positive numbers)
- Example: Z = {……, -3, -2, -1, 0, 1, 2, 3, …….}
E: Set of even natural numbers (numbers divisible by 2)
- Example: E = {2, 4, 6, 8, …….}
O: Set of odd natural numbers (numbers not divisible by 2)
- Example: O = {1, 3, 5, 7, 9, …….}

Rational and Irrational Numbers

Rational numbers can be expressed as a quotient of two integers (a fraction) with a non-zero denominator.
Irrational numbers cannot be expressed as a quotient of two integers (a fraction) with a non-zero denominator.
Examples:
- -12 is rational (can be written as -12/1)
- √25 is rational (can be simplified to 5/1)
- 0.09009000900009... is irrational (non-terminating and non-repeating)
- √3/4 is irrational (cannot be expressed as a quotient of two integers)
- √9/25 is rational (can be simplified to 3/5)

Set Builder Notation

A set-builder notation describes or defines the elements of a set instead of listing the elements.
Example: {x | x is a counting number less than 10} = {1, 2, 3, 4, 5, 6, 7, 8, 9}

Examples for Sets

A = {x | x ∈ N ∧ x > 7} = {8, 9, 10, …} (set of all natural numbers greater than 7)
Let C = {yellow, blue, red} (set of colors)
The cardinality of C is 3.

Standard Sets of Numbers (continued)

N: Set of natural numbers (1, 2, 3, …)
Z: Set of integers (…, -2, -1, 0, 1, 2, …)
Z+: Set of positive integers (1, 2, 3, …)
R: Set of real numbers (47.3, -12, π, …)
Q: Set of rational numbers (1.5, 2.6, -3.8, 15, …)

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Description

Test your understanding of sets and elements in discrete mathematics. Explore the concepts of finite and infinite sets, positive integers, set notation, and membership symbols.