Page 10 - DMTH403_ABSTRACT_ALGEBRA

Unit 1: Generating Sets

Some other sets that you may be familiar with are Notes

 a 
0
Q, the set of rational numbers =  a,b Z, b  
 b 
R, the set of real numbers

C, the set of complex numbers = (a+ib | a, b  R). (Here i   )
1.
Let us now see what subsets are.
Subsets: Consider the sets A = (1, 3, 4) and B = (1, 4). Here every element of B is also an element
of A. In such a case, that is, when every element of a set B is an element of a set A, we say that B
is a subset of A, and we write this as B  A.

It is obvious that if A is any set, then every element of A is certainly an element of A. So, for
every set A, A  A.

Also, for any set A,   A.
Now consider the set S = (1, 3, 5, 15) and T = (2, 3, 5, 7}. Is S  T? No, because not every element
of S is in T; for example, 1  S but 1  T. In this case we say that S is not a subset of T, and denote
it by S  T.

Note that if B is not a subset of A, there must be an element of B which is not an element of A. In
mathematical notation this can be written as ‘ x  B such that x’  A’.

We can now say that two sets A and B are equal (i.e., have precisely the same elements) if and
only if A  B and B  A.

Let us now look at some operations on sets. We will briefly discuss the operations of union,
intersection and complementation on sets.

Union: If A and B are subsets of a set S, we can collect the elements of both to get a new set. This
set is called their union. Formally, we define the union of A and B to be the set of all those
elements of S which are in A or in B. We denote the union of A and B by A  B. Thus,
A  B=(X  S | X  A or x  B).

For example, if A = {1, 2} and B = {4, 6, 7}, then A  B = {1, 2, 4, 6, 7}.
Again, if A = {l, 2, 3, 4) and B = (2, 4, 6, 8), then A  B = {l, 2, 3, 4, 6, 8). Observe that 2 and 4 are in
both A and B, but when we write A  B, we write these elements only once, in accordance with
Convention 2 given earlier.

Can you see that, for any set A, A U A = A?
Now we will extend the definition of union to define the union of more than two sets.
If A , A , A , ..........,A are k subsets of a set S, then their union A  A  . ......  A is the set of
k
k
1
2
2
1
3
elements which belong to at least one of these sets. That is,
A  A  ........  A = {x  S | x  A for some i = 1, 2 ,......, k).
1
i
k
2
k
The expression A  A  ........  A is often abbreviated to  A .
i
k
2
1
i 1

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