LESSON 2.3 Sets

Question 1

What is a set in mathematics?

Answer 1

A set is a well-defined collection of objects called elements.

Question 2

What does it mean for a collection to be well-defined?

Answer 2

A collection is well-defined if, for any given object, we can objectively decide whether it is or is not in the collection.

Question 3

What do we call an object that belongs to a given set?

Answer 3

Any object that belongs to a given set is called an element or a member of the set.

Question 4

Give an example of a well-defined set.

Answer 4

The collection of all letters in the English alphabet is a well-defined set.

Question 5

Why is the collection of all handsome guys not considered a set?

Answer 5

The collection of all handsome guys is not a set because “handsome” is a subjective term, and one cannot objectively identify if a given guy is handsome or not.

Question 6

How are sets commonly named in mathematics?

Answer 6

Upper case letters are usually used to name sets.

Question 7

What are the three common methods to describe a set?

Answer 7

A set can be described by (a) listing (roster) method, (b) set-builder notation, or (c) descriptive method.

Question 8

What is the roster method of describing a set?

Answer 8

The roster method describes a set by listing all its elements between braces and separated by commas. Each element is listed only once, and the arrangement is immaterial.

Example:
The set of months in a year that end with the letter ‘y’ can be represented by the roster {January, February, May, July}.

Question 9

What is set-builder notation?

Answer 9

Set-builder notation uses a variable, braces, and a vertical bar (|) to describe a set, usually when the elements are too many to list.

The set {2, 3, 4, 5, 6, 7, 8, 9} can be represented in set-builder notation as {x | x is an integer greater than 1 but less than 10}.

Question 10

What does it mean if a ∈ A?

Answer 10

If a ∈ A, it means that ( a ) is an element of the set ( A ).

Question 11

What symbol is used to denote the empty set?

Answer 11

The symbol ∅ sometimes just {} is used to denote the empty set.

Question 12

What is a unit set or singleton?

Answer 12

A unit set or singleton is a set that contains only one element.

Example:
The set of even prime numbers, which contains only the number 2, is a singleton set

Question 13

What does the symbol ℤ represent?

Answer 13

The symbol ℤ represents the set of integers, which includes all whole numbers: {… −4, −3, −2, −1, 0, 1, 2, 3, …}.

Question 14

What is the set of natural numbers denoted by?

Answer 14

The set of natural numbers (positive integers) is denoted by ℕ which is {1, 2, 3, 4, …}.

Question 15

What is the definition of rational numbers represented by ℚ?

Answer 15

The set of rational numbers is represented by ℚ = {a/b | a, b ∈ ℤ, b ≠ ∅}, where ( a ) and ( b ) are integers and ( b ) is not equal to zero.

Question 16

What does ℝ denote in mathematics?

Answer 16

The symbol ℝ denotes the set of real numbers.

Question 17

What is a finite set?

Answer 17

A set is finite if it is possible to list down all of its elements. The number of elements in a finite set is called the cardinality, denoted by n(A).

Question 18

What is the cardinality of a set?

Answer 18

The cardinality of a set is the number of elements it contains, denoted by n(A) for a set A. For example, the cardinality of the set of all letters in the English alphabet is 26.

Question 19

What is an infinite set?

Answer 19

A set is infinite if it is not possible to list all of its elements. An example is the set of all even integers, which goes on infinitely.

Question 20

What does it mean for a set A to be a subset of set B (A ⊆ B)?

Answer 20

A set A is a subset of set B if every element of A is also an element of B. This is written as A ⊆ B.

Question 21

Are sets always subsets of themselves?

Answer 21

Yes, every set is a subset of itself, i.e., A ⊆ A for any set A. Also, the empty set (∅) is a subset of every set.

Question 22

What does it mean when two sets A and B are equal?

Answer 22

Two sets A and B are equal (A = B) if every element of A is in B and every element of B is in A, which means A ⊆ B and B ⊆ A.

Question 23

Can two sets be equivalent without being equal?

Answer 23

Yes, two sets are equivalent if they have the same number of elements (n(A) = n(B)), but they do not have to contain the same elements. For example, A = {prime numbers less than 20} and B = {1, 2, 3, 4, 5, 6, 7, 8} are equivalent but not equal.

Question 24

What is the union of two sets A and B?

Answer 24

The union of two sets A and B, denoted A ∪ B, is the set of all elements that are in A, in B, or in both. In other words, it combines all the elements from both sets. It represents “A or B” (like disjunction).

Question 25

What is the intersection of two sets A and B?

Answer 25

The intersection of two sets A and B, denoted A ∩ B, is the set of all elements that are in both A and B. It contains only the elements shared by both sets. It represents the statement “A and B” (like conjunction).

Question 26

What is the relative complement of set B in set A?

Answer 26

The relative complement of B in A, denoted A \ B, is the set of all elements that are in A but not in B. { x ∈ A | x ∉ B }

Question 27

Give an example of a finite set and its cardinality.

Answer 27

The set of all letters in the English alphabet is finite, and its cardinality is 26.

Question 28

Give an example of an infinite set.

Answer 28

The set of all even integers is infinite because it cannot be completely listed.

Question 29

Can the empty set be a subset of any set?

Answer 29

Yes, the empty set (∅) is a subset of any set, since there are no elements in ∅ that could contradict the subset definition.

Question 30

What is the cardinality of the empty set?

Answer 30

The cardinality of the empty set is 0 because it contains no elements.