Theory:

 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}.