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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 .
We can read as 'A is a subset of B'.
Suppose and then .
If A is not a subset of B, then we can write .
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, . 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 and , 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 .
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, and .
Now let us write the elements and its combination of subsets.
The singleton subsets are and .
Therefore, the subsets of the set A are .
2. Consider the set with three elements .
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, and .
Now let us write the elements and its combination of subsets.
The singleton subsets are , and .
Let us write the subsets with two elements.
.
Therefore, the subsets of the set B are .