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Mathematics LibreTexts

10.1: Basics

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    An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a set.

    Set

    A set is a collection of distinct objects, called elements of the set

    A set can be defined by describing the contents, or by listing the elements of the set, enclosed in curly brackets.

    Example 1

    Some examples of sets defined by describing the contents:

    a) The set of all even numbers

    b) The set of all books written about travel to Chile

    Some examples of sets defined by listing the elements of the set:

    a) {1, 3, 9, 12}

    b) {red, orange, yellow, green, blue, indigo, purple}

    A set simply specifies the contents; order is not important. The set represented by {1, 2, 3} is equivalent to the set {3, 1, 2}.

    Notation

    Commonly, we will use a variable to represent a set, to make it easier to refer to that set later.

    The symbol ∊ means “is an element of”.

    A set that contains no elements, { }, is called the empty set and is notated ∅

    Example 2

    Let A = {1, 2, 3, 4}

    To notate that 2 is element of the set, we’d write 2 ∊ A

    Sometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris’s collection is a set, we can also say it is a subset of the larger set of all Madonna albums.

    Subset

    A subset of a set A is another set that contains only elements from the set A, but may not contain all the elements of A.

    If B is a subset of A, we write BA

    A proper subset is a subset that is not identical to the original set – it contains fewer elements.

    If B is a proper subset of A, we write BA

    Example 3

    Consider these three sets

    A = the set of all even numbers           B = {2, 4, 6}   C = {2, 3, 4, 6}

    Here BA since every element of B is also an even number, so is an element of A.

    More formally, we could say BA since if x B, then x A.

    It is also true that BC.

    C is not a subset of A, since C contains an element, 3, that is not contained in A

    Example 4

    Suppose a set contains the plays Much Ado About Nothing, MacBeth, and A Midsummer’s Night Dream. What is a larger set this might be a subset of?

    There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature.

    Try it Now 1

    The set A = {1, 3, 5}. What is a larger set this might be a subset of?

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