Elementary Abstract Algebra- Examples and Applications, 2019a

1.2 INTEGERS, RATIONAL NUMBERS, REAL NUMBERS 15
(a) For each of the properties (D),(E),(F) above, give (i) a specific example
for addition, using numbers, and (ii) a general statement for multiplication,
using variables. For example, for property (E) (the commutative
property) a specific example would be 3 + 5 = 5 + 3, and a general
statement would be x · y = y · x.
(b) Give a specific example that shows that subtraction is not commutative
.
(c) Give a specific example that shows that division is not associative.
♦
Exercise 1.2.3. Whichoftheabovepropertiesmustbeusedtoproveeach
of the following statements? (Note each statement may require more than
one property)
(a) (x + y)+(z + w) =(z + w)+(x + y)
(b) (x · y) · z =(z · x) · y
(c) (a · x + a · y)+a · z = a · ((x + y)+z)
(d) ((a · b) · c + b · c)+c · a = c · ((a + b)+a · b)
Note that the associative property allows us to write expressions without
putting in so many parentheses. So instead of writing (a + b)+c, wemay
simply write a + b + c. By the same reasoning, we can remove parentheses
from any expression that involves only addition, or any expression that
involves only multiplication: so for instance, (a · (b · c) · d) · e = a · b · c · d · e.
Using the associative and distributive property, it is possible to write any
arithmetic expression without parentheses. So for example, (a · b) · (c + d)
canbewrittenasa · b · c + a · b · d. (Remember that according to operator
precedence rules, multiplication is always performed before addition: thus
3 · 4 + 2 is evaluated by first taking 3 · 4 and then adding 2.)
♦
Exercise 1.2.4. Rewrite the following expressions without any parentheses,
using only the associative and distributive properties (don’t use commutative
in this exercise!)