Numbers - Session 6

In mathematics, a universal set means the largest set in a problem. It contains all the items that we are studying at that moment.Every smaller set we make must come from this universal set. You can imagine it like a big box that has everything we are talking about, and inside the box we can take out smaller groups. The universal set is usually written with the letter-U.

For example, if we are working with numbers from one to ten, then the universal set is the group of all the numbers from one to ten. From this main group, we can form smaller groups. One smaller group could be the even-numbers from one to ten, which are two, four, six, eight, and ten. Another smaller group could be the odd-numbers between one to ten, which are one, three, five, seven, and nine. Both of these smaller groups are part of the universal set because they all come from the numbers one to ten.

If we are studying shapes, then the universal set could include all shapes we are looking at, such as circle, square, triangle, and rectangle. From this universal set, we can make smaller groups. One group might be shapes with straight-sides, like square, rectangle, and triangle. Another group might be shapes with curved sides, like circle.

The universal set can also be very large. For example, if we say the universal set is all living things, then it will include humans, animals, plants, and trees. From this, we can make smaller sets, such as animals or plants. Another very large example is when we say the universal set is all natural numbers, which means the counting numbers starting from one and continuing without end. From this set, we can make smaller groups such as all even-numbers or all odd-numbers.

In Venn diagrams, the universal set is shown as a large rectangle. Inside the rectangle we draw circles for smaller sets. If the universal set is all students in a class, then one circle might be students who like football, another circle might be students who like cricket, and a third circle might be students who like swimming. The rectangle shows the whole class, including students who do not fit in any of the circles.

A subset means a smaller set that comes from a bigger set. If every item in one set is also found inside another set, then the first set is called a subset of the second set. This shows that subsets are always part of a larger group.For example, if our universal set is the numbers from one to ten, then the set of even-numbers between one to ten is a subset of the universal set, because all these numbers come from one to ten. In the same way, the set of odd-numbers between one to ten is also a subset of the universal set. Both of these smaller sets fit completely inside the bigger set.

When we draw Venn diagrams, we can indicate subsets as one circle completely inside another circle. For example, if the universal set is all fruits, and one smaller set is tropical fruits like mango and pineapple, then the circle of tropical fruits will fit completely inside the bigger circle of all fruits. This shows clearly that the smaller set is a subset of the larger one.

To indicate this in mathematics, we use a special symbol. We write it as A ⊆ B and read-out it as A is a subset of B. This means every element of set-A is also an element of set-B. For example, if A is the set of even-numbers such as two, four, and six, and B is the set one, two, three, four, five, and six, then we can write A ⊆ B. This shows that all the numbers in set-A are already found in set-B. Suppose set-A contains all odd-numbers, and set-B contains all prime numbers. Can you tell whether set-B is the subset of set-A or not?.

There are two special rules you should always remember. First, every set is a subset of itself. Second, the empty set is a subset of every set, because it has nothing inside it, so the rule is never broken.

Sometimes, a set is not a subset of another set. This happens when at least one element in the smaller set does not belong to the bigger set. In other words, if one item of the first set is missing from the second set, then the first set cannot be called a subset. For example, if set-A is one, two, and three, and set-B is two, three, and four, then A is not a subset of B, because the number one is in set-A but not in set-B. In mathematics, we express not a subset using the symbol that looks like a subset-sign with a small line through it. We write it as A ⊈ B, and it means A is not a subset of B.

An empty set is a set that has no elements inside it. It is also called the null set. This means it is a group, but the group is completely empty. There is nothing to count, nothing to list, and nothing inside. We represent empty set-By symbol ∅.

For example, if we talk about the set of whole numbers less than zero, that set will be empty. Whole numbers are zero, one, two, three, and so on. Since none of them are less than zero, the set has no members at all. Another example is the set of squares with three sides. A square always has four sides, so there cannot be a square with three sides. That means the set is empty.

The empty set is very useful in mathematics because sometimes we want to indicate that no answer or no object exists in a given situation. For instance, if we make a set of students in a class who are ten feet tall, the set will be empty because there are no students who are ten feet tall. If we make a set of prime numbers between eight and ten, the set will also be empty, because there is no prime number in that range.

Sometimes, we could get confused between an empty set-And a set that has the number zero in it. A set with the number zero inside, is not empty, because the number zero is actually one element in that set. An empty set has no elements at all. This is a very important difference to remember.

In diagrams, we can indicate the empty set-As a circle with nothing inside it. For example, if the universal set is all the students in a school, and we make a set of students who can fly-up like birds, that set will be empty. On the Venn diagram, it will appear as a circle, but the circle has no names inside it because no student has that ability.