Numbers - Session 5

Have you ever made a list of your favorite things, like your favorite fruits or games? In mathematics, we call such a list a set. A set is simply a group of things that we put together. These things are called elements of the set. For example, if we make a set of fruits, it can include an apple, a mango, and a banana. That means the apple, the mango, and the banana are elements of this fruit set.

Sometimes we want to indicate if something is inside a set or not. Instead of writing long sentences, we use short forms. If a number is in a set, we say it belongs to the set. If it is not in the set, we say it does not belong to the set. This makes our work easier and faster to understand.

The way of writing sets is like this. We use curly brackets. Curly brackets look like little curved arms. Inside the curly brackets, we write the elements of the set, separated by commas. For example, if we want to make a set of fruits that has apple, mango, and banana, we write it inside curly brackets as illustrated. This tells us very quickly what things belong to that set.

Now imagine you have a basket of toys. You want to tell your friend what toys are inside the basket. Instead of writing a long story, you can simply make a short list. This short way of writing is called set notation. A set is simply a group of things, and set notation is the special way we write that group.

For example, let us say your basket has a car, a ball, and a doll. We can say that set-A is the group of car, ball, and doll. We shall write the car, ball, and doll, separated by commas, inside curly brackets. If your basket had numbers instead, like one, two, and three, we would say that set-A is the group of numbers one, two, and three. So, we shall write one, two, and three, separated by commas, inside the curly brackets.

We can also make sets with numbers. If we want to make a set of the first three counting numbers, we write it as one, two, three inside the curly brackets. If we want to make a set of even-numbers less than ten represented by C, we write it as two, four, six, and eight, separated by commas, inside the curly brackets. See how neat and easy that looks.

Now let us talk about how a set can interact with another sets.A Venn diagram is a picture that helps us understand sets better. Instead of only writing sets with curly brackets, we can also draw them. In a Venn diagram, we usually draw circles. Each circle stands for one set, and the things inside that set are written inside the circle. If two sets have some things in common, the circles overlap, and we write the common things in the overlapping part.

Let us try an example. Suppose one set is the set of fruits including apple, mango, and banana. Another set is also the set of fruits including banana, orange, and mango. If we draw this as a Venn diagram, we make two circles. The first circle has apple, mango, and banana. The second circle has banana, orange, and mango. Now, mango and banana are in both sets, so they should be put in the middle where the two circles overlap. Apple stays only in the first circle, and orange stays only in the second circle. By looking at the picture, we can clearly see what belongs to each set and what they share.

Now let us try this with sets of natural numbers and prime numbers. Let us say, set-A is of natural numbers up to ten. Set-B is of prime numbers before 15. If you draw the Venn Diagram of these two sets, you will see that one, three, five, and seven are common numbers in both sets. So, they will be written in the overlapping section of both circles. Other numbers will be in their respective individual sections.

You can also represent common multiples of two numbers in form of Venn Diagrams. Let us say, multiples of two are two, four, six, eight, and ten. Let’s call this set as set-A. Now suppose, multiples of three are three, six, and nine. We can call this set has set-B.

Now represent the set-A and set-B in a Venn diagram. Next, let’s find if there is any common multiple in both sets. Six is the common multiple in both sets. We shall write six in overlapping section of both Venn diagrams. The remaining numbers are written in their respective Venn diagrams.

Venn diagrams can also represent three sets. For example, one set could be represented by letter-A that includes numbers such as two, four and six. Another set is represented by letter-B that includes numbers such as four, six, and eight. A third set is represented by letter-C which includes numbers such as six and ten.

If we draw three circles that overlap, the number six should be put in the middle where all three circles meet, because it is in all the sets. The number four should be put in the overlap between the first two circles, because it is common to them. The numbers two, eight, and ten will each be put in their own places. This way, the Venn diagram gives us a clear picture of how the sets connect.

Every set has things inside it. These things are called elements. The number of elements in a set simply means how many things are in that set. We count them the same way we count objects around us. For example, if we make a set of fruits with apple, mango, and banana, then the number of elements is three because there are three fruits in the group. If we make a set of numbers with two, four, six, eight, and ten, then the number of elements is five because there are five numbers in the group.

Now, how do we represent this in mathematics? We use the letter-n in front of the name of set.So, we write n(A) = 3 to indicate that Set-A has three elements. We write n(B) = 5 to indicate that Set B has five elements.