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Unit 1: Generating Sets

5. Let a, b, c be non-zero integers. Then Notes
(a) a|0, ±1|a, ±a|a.

(b) a|b  ac|bc
(c) a|b and b|c  a|c
(d) a|b and b|a  a = ±b
(e) c|a and c|b  c | (ax + by)  x, y  Z.

6. If p is a prime and p|ab, then show that p|a or p|b.
7. If p is a prime and p|a a .....a , then show that p|a for some i = 1,....,n.
1 2
i
n
Answers: Self Assessment
1. A is consisted of elements: 3, {4, 5} and 8, so {4, 5} is an element of A

2. {1,2,3, ..., 1000} is finite, because it is consisted of final number of elements.
3. Set B is not an empty set because it contains one element. The only element of the set B is
zero. B = {0}
4. a E is true, because a  D and D  E means that every element from D is contained in E.

5. The correct answer is E, because E consists of even numbers as elements and the intersection
of sets S and B is a null set.
6. A  B is the correct answer because A is a superset of B.

7. The correct answer is {1, 2, 3} because all subsets of {1, 2, 3} are , {1}, {2}, {3}, {1, 2}, {1, 3},
{2, 3}, {1, 2, 3}.

1.9 Further Readings

Books Dan Saracino: Abstract Algebra; A First Course.
Mitchell and Mitchell: An Introduction to Abstract Algebra.
John B. Fraleigh: An Introduction to Abstract Algebra (Relevant Portion).

Online links www.jmilne.org/math/CourseNotes/
www.math.niu.edu
www.maths.tcd.ie/

archives.math.utk.edu

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