About Lesson
Subsets:
If P and Q are two sets given in such a way that every element of P is in Q, then we say P is a subset of Q and we write it as P ⊆ Q
Therefore if P ⊆ Q and x belong to P then x belongs to Q
- Every set is a subset of itself.
i.e. P ⊆ P, Q ⊆ Q, etc. - Empty set is a subset of every set
i.e. ϕ ⊆ P, ϕ ⊆ Q - If P ⊆ Q and Q ⊆ P, then P = Q
Example:
If P = {-9,13,6},
then,
Subsets of P= ϕ, {-9}, {13}, {6}, {-9,13}, {13,6}, {6,-9}, {-9,13,6}
Proper Subset:
Let P be any set and let Q be any non-empty subset. Then P is called a proper subset of Q, if and only if every element of P is in Q, and there exists at least one element in Q which is not there in P.
- i.e. if P ⊆ Q and P and Q are not equal, then A will be a proper subset of Q
- Please note that ϕ has no proper subset
- A set containing n elements has (2^n – 1) proper subsets.
- i.e. if P = {1, 2, 3, 4}, then the number of proper subsets is (2^4 – 1) = 15
Power Set:
The set of all possible subsets of a set A is called the power set of A, denoted by P(A). If A contains n elements, then P(A) = 2^n sets.
i.e. if A = {1, 2}, then P(A) = 2^2 = 4
- Empty set is a subset of every set
- So in this case the subsets are {1}, {2}, {1, 2} & ϕ