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Let A and B are two sets. If every element of A is also an element of B, then A is a subset of B. It is denoted as AB.
We can read AB as 'A is a subset of B'.
 
Suppose AB and {1}A then {1}B.
 
If A is not a subset of B, then we can write AB.
 
Important!
  • If A is a subset of B, the number of elements in the set A must be less than or equal to the number of elements in the set B. That is, n(A)n(B). Since every element of A is also an element of B, the set B must have at least as many elements as A, thus n(A) ≤ n(B). It can be concluded that if A is a subset of B, then the cardinal number of A must be less than the cardinal number of B.
  • If AB and  BA, then A = B.
  • An empty set is a subset of every set.
  • Every set is a subset of itself.
Example:
1. Consider the set with two elements A={1,2}.
 
By the concept, 'Empty set is a subset of every set' and 'Every set is a subset of itself' the two obvious subsets of the set are the empty set and the whole set itself.
 
That is, or{} and {1,2}.
 
Now let us write the elements and its combination of subsets.
 
The singleton subsets are {1} and {2}.
 
Therefore, the subsets of the set A are {},{1},{2}and{1,2}.
 
 
2. Consider the set with three elements B={a,b,c}.
 
By the concept, 'Empty set is a subset of every set' and 'Every set is a subset of itself' the two obvious subsets of the set are the empty set and the whole set itself.
 
That is, or{} and {a,b,c}.
 
Now let us write the elements and its combination of subsets.
 
The singleton subsets are {a}, {b} and {c}.
 
Let us write the subsets with two elements.
 
{a,b},{a,c},{b,c}.
 
Therefore, the subsets of the set B are {},{a},{b},{c},{a,b},{a,c},{b,c}.