Difference between proper and improper subsets . Does the same apply for super sets . And any others ?
Dear Student
Proper and improper subsets: If A is a subset of B and A ≠ B, then A is a proper subset of B. We write this as A ⊂ B.
The null set ϕ is subset of every set and every set is subset of itself, i.e., ϕ ⊂ A and A ⊆ A for every set A. They are called improper subsets of A. Thus every non - empty set has two improper subsets. It should be noted that ϕ has only one subset ϕ which is improper.
All other subsets of A are called its proper subsets. Thus, if A ⊂ B, A ≠ B , A ≠ ϕ , then A is said to be proper subset of B.
Example : Let A = {1, 2}. Then A has ϕ ; {1}, {2}, {1, 2} as its subsets out of which ϕ and {1, 2} are improper and {1} and {2} are proper subsets.
Proper and improper subsets: If A is a subset of B and A ≠ B, then A is a proper subset of B. We write this as A ⊂ B.
The null set ϕ is subset of every set and every set is subset of itself, i.e., ϕ ⊂ A and A ⊆ A for every set A. They are called improper subsets of A. Thus every non - empty set has two improper subsets. It should be noted that ϕ has only one subset ϕ which is improper.
All other subsets of A are called its proper subsets. Thus, if A ⊂ B, A ≠ B , A ≠ ϕ , then A is said to be proper subset of B.
Example : Let A = {1, 2}. Then A has ϕ ; {1}, {2}, {1, 2} as its subsets out of which ϕ and {1, 2} are improper and {1} and {2} are proper subsets.
Whenever a set A is a subset of set B, we say the B is a superset of A and we write, B ⊇ A.
Symbol ⊇ is used to denote ‘is a super set of’
For Example:
A = {a, e, i, o, u}
B = {a, b, c, ............., z}
Here A ⊆ B i.e., A is a subset of B but B ⊇ A i.e., B is a super set of A
Regards