Learning objectives
By the end of this lesson, you should be able to;
Let us start by learning about sets. What are sets? A set is simply defined as a collection.
How do we create sets? We create sets by specifying a common property among things, and then gather up everything that has this common property. For example, we can have a set of things you wear. This includes shirt, hat, trouser and jacket. This is called a set. Another example of a set is the types of fingers. This set is made up of the thumb, index, middle, ring, and the pinky finger. Therefore, a set is just a group of things brought together by a certain common property.
Notation of sets
To denote sets, we simply listing each member or element and separating them by a comma. We also use braces to enclose a set. These braces are sometimes called set brackets. For example, {thumb, index, middle, ring, and pinky}, and {shirt, hat, trouser and jacket} are sets.
Numerical sets
We also have sets in mathematics. When defining sets, we just need to specify a common characteristic. For example, we can have a set of even numbers between 0 and 10 {2, 4, 6, 8}, a set of odd numbers between 0 and 10 {1, 3, 5, 7, 9}, and a set of prime numbers between 0 and 10 {2, 3, 5, 7}.
Importance of sets
Sets are an important property of mathematics. The application of sets in mathematics includes in abstract algebra, graph theory, linear algebra, and binary operations. Now, let us move on to a new concept called operations.
Operations
Since we have already learned about sets and their elements, let us look at how to work with them. The process of combining more than one set of elements to produce other elements is called operation. It can be simply put as; an operation combines elements of a set.
Binary operations
A binary operation is similar to an operation but it involves combining only two elements into 1. Any operation that involves combining more than two elements is not a binary operation. The following are examples of common binary operations, 5 + 3 = 8. 4 x 3 = 12. 4 – 4 = 0. From these examples, we see that two numbers combine and become one. Note that, even for two numbers that are similar, but combine to form one, it is also considered to be a binary operation.
Well defined operators
In binary operations, the operators or elements must be well defined. What do we mean by well defined? A well-defined binary operation is an operation that has one answer only. For example, in the binary operation 5 + 3, there is only one answer to expect 8. However, not all operations are like this. Take for instance square roots. The operation x2 = 25 has two answers, 5 and -5. With well-defined operators, there exists only one possible answer.
It is also important to note that we sometimes use the symbol * to denote an operation.
A combination of a set and an operation forms a group.
