What is the difference between a subset and a proper subset?
Solution:
In set theory, a proper subset of a set A is a subset of A that cannot be equal to A. In other words, if B is a proper subset of A, then all elements of B are in A but A contains at least one element that is not in B.
Subset:
If A and B are sets and every element of A is also an element of B, then:
A s a subset of B, denoted by A ⊆ B (or equivalently, B is a superset of A, denoted by B ⊇ A).
For example, A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
A = {1, 2, 3} and B = {1, 2, 3}
Here, A is a subset of B, or we can say that B is the superset of A.
Proper Subset:
If A is a subset of B, but A is not equal to B (that is, there exists at least one element of B which is not an element of A), then A is also a proper (or strict) subset of B; this is written as A ⊊ B (or) A ⊂ B.
For example A = {1, 2, 3} and B = {1, 2, 3, 4}.
Clearly, A is not equal to B and element 4 belongs to set B but is absent in set A, so we have one element in set B which is not an element of set A. Thus, A can be called a proper subset of B.
Hence, the differences between subset and proper subset can be summarized in the table below:
| Subset | Proper Subset |
|---|---|
| If A is a subset of B, we can write it as A ⊆ B. | If A is a proper subset of B, we can write it as A ⊊ B (or) A ⊂ B. |
| If A ⊆ B, then A is a subset of B and A may or may not be equal to B. | If A ⊂ B, then A is a subset of B but A is NOT equal to B. |
| (i) {1, 2, 3} is a subset of {1, 2, 3}. (ii) {1, 2, 3} is a subset of {1, 2, 3, 4} |
(i) {1, 2, 3} is NOT a proper subset of {1, 2, 3}. (ii) {1, 2, 3} is a proper subset of {1, 2, 3, 4} |
What is the difference between a subset and a proper subset?
Summary:
A subset of a set A can be equal to set A but a proper subset of a set A can never be equal to set A.
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