(a) Define power set, null set, …

Any Set that does not contain any element is called the empty or null or void set. The symbol used to represent an empty set is – {} or φ. Examples:

• Let A = {x : 9 < x < 10, x is a natural number} will be a null set because there is NO natural number between numbers 9 and 10. Therefore, A = {} or φ
• Let W = {d: d > 8, d is the number of days in a week} will also be a void set because there are only 7 days in a week.

A universal set is a set which contains all the elements or objects of other sets, including its own elements. It is usually denoted by the symbol ‘U’.

Suppose Set A consist of all even numbers such that, A = {2, 4, 6, 8, 10, …} and set B consist of all odd numbers, such that, B = {1, 3, 5, 7, 9, …}. The universal set U consist of all natural numbers, such that, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,….}. Therefore, as we know, all the even and odd numbers are a part of natural numbers. Therefore, Set U has all the elements of Set A and Set B.

The power set is a set which includes all the subsets including the empty set and the original set itself. It is also a type of sets. If set A = {x,y,z} is a set, then all its subsets {x}, {y}, {z}, {x,y}, {y,z}, {x,z}, {x,y,z} and {} are the elements of powerset, such as:

Power set of A, P(A) = {x}, {y}, {z}, {x,y}, {y,z}, {x,z}, {x,y,z} and {}

Where P(A) denotes the powerset.