A collection of well-defined objects is called a set.
Every object has some properties naturally and the properties of one object is compared with one or more objects. If either one or more objects have same property or properties, then the collection of such objects is called a set.
Study the following examples to understand how to define and form a set mathematically.
There are three apples in this example. Two are red apples and one is a green apple.
If you consider, apple is the actual property to form a set and don’t want to consider the color factor, then the collection of three of them is a set.
On the basis of the colour, the collection of them is not a set. In this case, two sets are possibly formed. The collection of two red apples forms a set and the collection of one green apple form another set.
There are some footballs in a football stadium and the total footballs is $8$.
All $8$ balls are footballs. Hence, the collection of all eight footballs is a set.
If we consider the size of the footballs, the collection of all eight footballs is not a set because three footballs are in actual size and the other five footballs are in small size. Therefore, the collection $3$ footballs is a set and the collection $5$ footballs is another set.
Observe the both cars. The properties such as colour, model and everything are same. Hence, the collection of both cars is a set.
The above examples give you an idea about defining a set mathematically according to set theory.
Remember one thing while defining a set; if you want to define a set on the basis of either one or more properties, the objects should have same property or properties commonly. Otherwise, the objects cannot form a set mathematically.