Set theory

Question

What is the difference between Subset and Proper subset?

Can we say subset is also a proper subset in some cases?

Comments

let A={1,2,3} than B={1,2} is proper subset of A and C={1,2,3} is a subset of A. yes we can say that because a proper subset is a subset of A which is not equal to A. 

https://en.wikipedia.org/wiki/Subset it says where two sets may or  maynot coincide . and moreover it says about every element of A is in B but not every element of B is in A.

Answer 1

only difference is that

Subset of a given set might contain all the element of given set BUT proper subset should not contain all element

eg: 

  The set {2,3,5,7} is a subset of {2,3,5,7}.

   The set {2,3,5,7} is NOT a proper subset of {2,3,5,7}.

   The set {2,3,5} is a proper subset of {2,3,5,7}.

   The set {NULL} is a proper subset of {2,3,5,7}.

The set {NULL} is a subset of {2,3,5,7}.

You have the answer in front of you. If A has cardinality n, then the number of subsets is 2ⁿ and the number of proper subsets is 2ⁿ−1, because the only set we have to "throw out" is A itself in order to get all the proper subsets.

 

Answer 2

If you have a subset $A\subset B$, $A$ is called a proper subset if  $  A \neq B$.

Therefore, every proper subset is also a subset, but every subset is not a proper subset.

Comments

A = 1, 3, 5, 7, 9

B = 1, 2,3,4,5,6,7,8,9,10

Here, we say that A is subset of B, as every elemrel of A is in B.

Hence A is Subset of B.

Now by definition of Proper Subset, as B contains 4 and A does not hence A and B are not equal sets, hence A is proper subset of B.

Here we can show that, A which is a subset is also a proper subset.

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A subset can be a proper subset, but all subsets are not proper subsets.

However, all proper subsets are subsets.