Subsets

If every member of a set  A is also a member of a set B, then set A is a subset of set B. It is denoted as A⊂B. For example, {14, 49, 77}⊂{n∈N|n is a multiple of 7}.

proper-subset

Let us see another example:

  1. Let C={n∈N|n is a multiple of 6}, D={n∈N|n is a multiple of 2 and 3}. Show that C⊂D.

Solution:  A number is a multiple of 6 if it satisfies the equation n=6m for some number m. We can write this as n=2x(3m) and n=3x(2m), which proves that n is a multiple of 2 and 3. So thus, all the elements of set C are also the elements of set D.

So, we conclude that C⊂D.

2. If A={2,4,8} and B={m|m is an even number}, then prove that A⊂B.

Solution: We know that A={2,4,8} and B={0,2,4,6,8,10,….}. Now, from this statement we can see that set B consists of all the even numbers which include the numbers 2, 4 and 8. These numbers are the element of set A. So, all the elements of set A are also the elements of set B.

So, we conclude that A⊂B.

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