Power Set Universal Set (Set theory part 6 NCERT)


Power Set

Consider the set {1, 2}. Let us write down all the subsets of the set {1, 2}. We know that φ is a subset of every set . So, φ is a subset of {1, 2}. We see that {1} and {2}are also subsets of {1, 2}. Also, we know that every set is a subset of itself. So, { 1, 2 } is a subset of {1, 2}. Thus, the set { 1, 2 } has, in all, four subsets, viz. φ, { 1 }, { 2 } and { 1, 2 }. The set of all these subsets is called the power set of { 1, 2 }. 

Definition 5 The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set. 

Thus, as in above, if A = { 1, 2 }, then 
P( A ) = { φ,{ 1 }, { 2 }, { 1,2 }} 
Also, note that n [ P (A) ] = 4 = 22 
In general, if A is a set with n(A) = m, then it can be shown that 
n [ P(A)] = 2m.

Universal Set

Usually, in a particular context, we have to deal with the elements and subsets of a basic set which is relevant to that particular context. For example, while studying the system of numbers, we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth. This basic set is called the “Universal Set”.

The universal set is usually denoted by U, and all its subsets by the letters A, B, C, etc.

For example, for the set of all integers, the universal set can be the set of rational numbers or, for that matter, the set R of real numbers.

For another example, in human population studies, the universal set consists of all the people in the world.