Page 11 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes If  is a collection of subsets of a set S, then we can define the union of all members of by

 A = (x  S | X " A for some A  ).
A
Now let us look at another way of obtaining a new set from two or more given sets.
Intersection: If A and B are two subsets of a set S, we can collect the elements that are common to
both A and B. We call this set the intersection of A, and B (denoted by A  B. So,
A  B = { x  S | X " A and x  B } .

Thus, if P = {1, 2, 3, 4} and Q = {2, 4, 6, 8}, then P  Q = {2, 4},
Can you see that, for any set A, A  A = A?

Now suppose A = {1, 2) and B = (4, 6, 7). Then what is A  B? We observe that, in this case, A and
B have no common elements, and so A  B = , the empty set.
When the intersection of two sets is , we say that the two sets are disjoint (or mutually disjoint).
For example, the sets (1, 4) and (0, 5, 7, 14) are disjoint.
The definition of intersection can be extended to any number of sets. Thus, the intersection of k
subsets A , A ,....., A of a set S is
1
2
k
A,  A  .........  A, = { x E S | x E A for each i = 1, 2,........, k }.
2 i
k
We can shorten the expression A  A  .........  A to  A . i
k
2
1
i 1

In general, if  is a collection of subsets of a set S, then we can define the intersection of all the
members of by
 A = { X  S | X  A V A   } [ V denotes ‘forever’]
A P

Apart from the operations of unions and intersections, there is another operation on sets, namely,
the operation of taking differences.
Differences: Consider the sets A = { 1, 2, 3] and B = [2, 3, 4]. Now the set of all elements of A that
are not in B is {1}. We call this set the difference A \ B. Similarly, the difference B \A is the set of
elements of B that are not in A, that is, {4}.

Thus, for any two subsets A and B of a set S,
A\B = { x  S | x A and x  B }
When we are working with elements and subsets of a single set X, we say that the set X is the
universal set. Suppose X is the universal set and A  X. Then the set of all elements of X which are
not in A is called the complement of A and is denoted by A’, A or X \ A.
C
Thus,
A = {x  X | x  A }.
c
For example, if X = [a, b, p, q , r) and A = {a, p, q], then A = (b, r).
c
1.2 Cartesian Product

An interesting set that can be formed from two given sets is their Cartesian product, named after
the French philosopher and mathematician Rene Descartes (1596 - 1650). He also invented the
Cartesian co-ordinate system.

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