Page 25 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 1.6 Summary

In this unit we have covered the following points:

Some properties of sets and subsets.

The union, intersection, difference and complements of sets.

The Cartesian product of Sets.

Relations in general, and equivalence relations in particular.

The definition of a function, a 1-1 function, an unto function and a bijective function.

The composition of functions.

The well-ordering principle, which states that every subset of N has a least element.

The principle of finite induction, which states that : If P(n) is a statement about some n  N

such that
(i) P(1) is true, and
(ii) if P(k) is true for some k  N, then P(k + l) is true,

then P(n) is true for every n  N.
The principle of finite induction can also be stated as:

If P(n) is a statement about some n  N such that
(i) P(l) is true, and

(ii) if P(m) is true for every positive integer m < k, then P(k) is true,
then P(n) is true for every n  N.
Note that the well-ordering principle is equivalent to the principle of finite induction.
Properties of divisibility in Z, like the division algorithm and unique prime factorisation.

1.7 Keywords

Empty Set: A set with no element in it is called the empty set, and is denoted by the Greek letter
 (phi). For example, the set of all natural numbers less than 1 is .
Roster Method: It is sometimes used to list the elements of a large set also. In this case we may
not want to list all the elements of the set.
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.
1.8 Review Questions

1. Let C = {1, 2, 3, 4} and D = {1, 3, 5, 7, 9}. How many elements does the set C  D contain?
How many elements does the set CD contain?

2. Let U = {1, 2, 3... 8, 9}, B = {1, 3, 5, and 7} and C = {2, 3, 4, 5, 6}. How many elements does the
set (B  C)’ contain? How many elements does the set (C – B)’ contain?
3. Let S = {a, b}. How many elements does the power set 2 contain?
S
4. Let S = {1, 2, 3}. How many subsets does S contain?

18 LOVELY PROFESSIONAL UNIVERSITY