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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
- 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?

- {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

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

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

- 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 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.

- 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 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.

- 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 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.

- , which is true.
- Also because {1,2,3,…} contains elements that are not members of {2,4,6,…}.

Exercise 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.

- Find all the subsets of {1,2,3}
- If a set has five elements, how many subsets will it have?
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