Comparing sets
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Comparing sets. Chapter 2 Sec 2. Set equality. One of the fundamental things we need to know about two sets is when do we consider them to be the same. Def

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Comparing sets

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Comparing sets

Comparing sets

Chapter 2 Sec 2


Set equality

Set equality

  • One of the fundamental things we need to know about two sets is when do we consider them to be the same.

  • Def

    • Two sets A and B are equal if they have exactly the same members. In this case, we write A = B. If A and B are not equal, we write A ≠ B.


Example are the sets equal

Example, are the sets equal?

  • {Socrates, Shakespeare, Armstrong} = {Armstrong, Socrates, Shakespeare}

  • A={x:x is a citizen of the US} and B={y:y was born in the US}


Subsets

Subsets

  • Another way we compare sets is to determine whether one set is part of another set.


Definition

Definition

  • The set A is a subset of the set B if every element of A is also an element of B. We indicate this relationship by writing . If A is not a subset of B, then we write


Comparing sets

  • In order to show that , we must show that every element of A also occurs as an element of B. To show that A is not a subset of B, all we have to do is find one element of A that is not in B.


Identifying subsets

Identifying subsets

  • Determine whether either set is a subset of the other.

  • A ={2, 5, 6} and B ={1,2, 5, 6}

    • Every member of A is in B, therefore we can write .

    • But, there is an element of B that is not in A,


Proper subset

Proper Subset

  • The set A is a proper subset of the set B if but A ≠ B.

  • We write this as .

  • If A is not a proper subset of B, then we write


Example

Example

  • , which is true.

  • Also because {1,2,3,…} contains elements that are not members of {2,4,6,…}.


Exercise

Exercise

  • Find all the subsets of {1,2,3}

  • If a set has five elements, how many subsets will it have?

    • 25


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