Set Operations

A set is a group of things, called members or elements, which are represented in one of two ways.

1. Roster Notation: The elements are listed inside set brackets with commas between each of the elements.
2. Set-Builder Notation: The same set brackets are listed, but in this case, a variable or variable expression is shown followed by either a colon or a vertical line and then a list of the necessary characteristics for members of the set.

Thus, we could write:  $\lbrace1,2,3\rbrace=\lbrace x\mid\:x\:is\:a\:natural\:number\:and\:x\:is\:less\:than\:4\rbrace$

Frequently in mathematics sets contain numbers. When you have several sets, all of which have elements that belong to the same larger set, the large set from which the elements of the smaller set belong is called the universal set and is denoted $U$.

For each set $A$, there is an associated other set, called the complement of $A$, or $A'$ (read "A prime"), which contains all elements that are in the universal set and that are not in $A$.

We can also perform several operations with sets, the most common of these are unions and intersections. Please watch the following video explaining what these operations mean, and how to perform them:

Sets can contain any number of elements, and the number of elements within a set is called the sets cardinality. The set with no elements, usually called either the empty set or the null set, can be written as $\phi$ or $\lbrace\rbrace$.

Sets that contain a specified number of elements are called finite sets. An example of a finite set would be $\lbrace yellow,\:red,\:blue\rbrace$. This set has 3 elements, so it has a cardinality of 3.

Infinite sets contain $\infty$ elements. For example the set of integers,$\lbrace...-3,-2,-1,0,1,2,3,...\rbrace$ , would be an infinite set, as would the set of natural numbers, $\lbrace1,2,3,...\rbrace$.

The symbol $\in$ means "Is an element of". Thus $3\in A$ would mean that 3 is an element contained in set $A$. If instead 3 is not an element of set $A$, we would instead write $3\notin A$, meaning 3 is not an element of $A.$

When all of the elements in set $A$ are also elements in set $B$, then we can say that $A$ is a subset of $B.$Here is a short video that introduces the concept of subset, proper subset, and superset.

At this point we have finished the material for section R2. Please go to the menu to your left and hit on "MyLab and Mastering". If this is your first time here, then you will need to pay to access this site, but once you do you should be able to complete by Wednesday morning at 9am the first two assignments posted there, or found on the next page of this module, namely please complete:

1. Orientation Homework
2. Homework R2