Welcome to our introduction to sets. A set is a well-defined collection of distinct objects. Think of it as a container that holds unique items. For example, we can have a set of vowels containing a, e, i, o, u, or a set of even numbers like two, four, six, eight. Each object in a set is called an element, and no element can appear more than once in the same set.
Now let's learn about set notation. There are several ways to represent sets. The roster method lists all elements inside curly braces, like A equals one, two, three, four, five. Set-builder notation describes the properties of elements, such as B equals the set of all x where x is even and x is less than or equal to ten. Interval notation is used for continuous ranges, like C equals zero to five, representing all real numbers between zero and five inclusive.
Let's explore relationships between sets. A subset relationship, denoted by the symbol contained in, means that every element of set A is also in set B. A proper subset means A is a subset of B, but A is not equal to B. Set equality means two sets have exactly the same elements. In our diagram, set A with elements one and two is a proper subset of set B, which contains one, two, three, four, and five.
Now let's learn about set operations. The union of two sets A and B, denoted A union B, contains all elements that are in A or B or both. The intersection of A and B contains only the elements that are common to both sets. The difference A minus B contains elements that are in A but not in B. These operations are fundamental tools for working with sets and are visualized using Venn diagrams like the one shown here.
To summarize what we've learned about sets: Sets are well-defined collections of distinct objects that can be represented in multiple ways. We explored set relationships like subsets and equality, and learned about fundamental operations including union, intersection, and difference. These concepts form the foundation of modern mathematics and are essential tools for organizing and analyzing data in various fields.