## Introduction to Sets## DescriptionThis is an introductory article to the theory of sets. It is meant to be skipped by those who are experts in sets. If you don’t know whether you should be reading the article or not, please scroll down and take a test that may help you decide on the quantity of attention and respect that you should be giving to this page. ## Introduction
Sets are collections of objects. Here are three examples:
In some cases it is possible to list all elements of the set. One such example is the set \( S \) of all two-digit integers that are divisible by \( 17 \). We can write \[ S=\{17,34,51,68,85\}.\] The order in which we list the elements does not matter. This means that \( \{17,34,51,68,85\} \) is precisely the same set as \( \{34,51,17,68,85\} \). Also, repetitions of elements do not affect sets. The set \( \{17 \), \( 17 \), \( 34 \), \( 34 \), \( 51 \), \( 68 \), \( 85\} \) is the same old \( S \). Usually we denote sets by capital letters from the alphabet. ## Elements and Subsets
If we want to emphasize that something is an
A set \( Q \) is a is written as \( Q\subseteq S \). Example: \[ \{34,68\}\subseteq S.\]## Union, intersection, and differenceTheunion \( A\cup B \) of two sets \( A \) and \( B \) is a set containing all elements that are in either \( A \) or \( B \). The intersection \( A\cap B \) is the set containing elements that are in both \( A \) and \( B \).
Let \( S=\{17,34,51,68,85\} \) and \( T=\{5, 7, 17, 71, 85\} \) be two sets. Then \[ S\cup T=\{5,7,17,34,51,71,58,85\}\;\;\mbox{and}\;\;\; S\cap T=\{17,85\}.\] The difference \( A\setminus B \) is the set containing those elements of \( A \) that are not in \( B \).
In the case of previously defined \( S \) and \( T \) we have \[ S\setminus T=\{34,51,68\}.\]
## Complement of a setIf we know that all elements we are dealing with are elements of a given fixed set \( U \), then we have a special notion for \( U\setminus S \). We write \[ S^C=U\setminus S\] and say \( S^C \) is a complement of \( S \).Then it is easy to prove the following identities: \[ (A\cup B)^C=A^C\cap B^C;\] \[ (A\cap B)^C=A^C\cup B^C;\] \[ A\setminus B=A\cap B^C.\] The above identities are called de Morgan’s laws. ## Other ways to write setsSo far we described sets by listing all of their elements. We want to study those sets where this cannot be done. We studied the set \( S \) of all two-digit positive integers that are divisible by 17. Consider now the set \( D \) ofall positive integers that are divisible by \( 17 \). We can’t list all elements of \( D \). However, we can write math propositions involving this set \( D \). Examples are:
\[ 85\in D\;\;\; 170\in D\;\;\; 710\not\in D\;\;\; D\cap S=S.\]
Although we don’t have the list of all elements of \( D \), we are able to guarantee that all the above propositions are correct.
If there is a property that uniquely defines the set we use the following syntax involving colon \( : \) to define our set \( D \)
\[ D=\{z: \;z\mbox{ is a positive integer that is divisible by }17\}.\]
Some books use vertical line \( | \) instead of colon, but you don’t want to do that, because in problem solving vertical line is so precious as it is used for divisibility (we write \( 17|85 \)). In some cultures, the symbol ## Cartesian product of sets
The cartesian product \( A\times B \) of sets \( A \) and \( B \) is the set of ordered pairs whose first component is from \( A \) and the second component is from \( B \). For example, if \( A=\{1,2,3,4\} \) and \( B=\{3, a, 9\} \), then their cartesian product is:
\[ A\times B=\{(1,3), (1, a), (1,9), (2,3), (2,a), (2,9), (3,3), (3,a), (3,9), (4,3), (4,a), (4,9)\}.\]
We use the notation \( A^2 \) for \( A\times A \). |

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