1
Problems and Comments For Section 2
Problems: 2.1, 2.5, 2.7, 2.8
Problem 2.13 requires some thought. It is optional.
A product term is dened recursively as follows:
1. p = a is a product where a is any element of the group G.
2. If p1 and p2 are product
Problems and Comments For Section 0
Problems: 0.7, 0.18; 0.16
Not all problems on Induction require induction. 0.16 is better proven on the basis of
the division algorithm (p. 20, Lemma 2.1). Of course this algorithm is known to you
from elementary school
Problems and Comments For Section 3
Problems: 3.1, 3.4, 3.9, 3.10, 3.11, 3.12
Comments: For the multiplicative monoid of n n matrices one has that a left-inverse of a
matrix A is automatically a right-inverse, thus an inverse.
This is proven in linear alg
Problems and Comments For Section 4
Problems: 4.4, 4.9, 4.10, 4.15, 4.17,
Comments: For integers m and n we define that m divides n, written m|n, in case that
there is some k such that k m n. We have,
1. 1|n; n|0.
2. n|m and m|n if and only if n m.
3. If
Problems and Comments for Section 5
Problems: 5.1, 5.2, 5.7, 5.18 (a), 5.19, 5.20, 5.21
Comments: For a given algebra A, f i iI , a subset C is called closed if it is closed
under all the operations: If for example f i is a binary operation, say , and if
Problems and Comments for Section 17, 18, and 21
Problems: 17.6, 17.7, 18.1 (a), (b), (c), 21.11, 21.12
Comments (and synopsis for these sections): You should read 17 and 18
simultaneously. You may stop reading section 18 after the examples for Theorem
18