The concept of "Sets" is a fundamental part of modern mathematics. It serves as the basic building block for advanced chapters like Relations, Functions, Probability, and Geometry. In this detailed guide, we will cover everything from basic definitions to practical formulas.
Here, "well-defined" means that there should be no ambiguity regarding whether a particular object belongs to the collection or not. The answer must remain the same regardless of who is asked.
There are two primary methods to represent a set:
In this method, all elements of a set are listed inside braces { } and are separated by commas. (Order doesn't matter, no repetitions).
Instead of listing every element, we write a common property or rule that holds true for all elements of the set. Example: {x : x is a natural number < 5}.
- Empty / Null Set (∅): A set which does not contain any element.
- Singleton Set: A set containing exactly one element. Example: {5}
- Finite Set: A set where the process of counting elements comes to an end.
- Infinite Set: A set whose elements cannot be counted (infinity).
- Equal Sets: Two sets A and B are equal (A = B) if they have exactly the same elements.
If every element of set A is also an element of set B, then A is a Subset of B. Denoted by A ⊆ B.
Union means combining all elements. Intersection means taking only the common elements.
Elements that belong to A but do NOT belong to B. Simply removing the B portion from A.
The complement of A is the set of all elements of the Universal set which are not the elements of A.
A' = U - A
- (A ∪ B)' = A' ∩ B'
- (A ∩ B)' = A' ∪ B'
These formulas are essential for solving word problems (like survey questions):
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
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