1. MaryJane R. De Ade
BATSTATEU - JPLPC Malvar Campus

2. Notations
ℝ set of real numbers
ℂ set of complex numbers
ℚ set of rational numbers
ℚ′
set of irrational numbers
𝕎 set of whole numbers
ℤ set of integers

3. ℤ+
set of positive integers
ℤ− set of negative integers
𝐴, 𝐵 sets
𝑎, 𝑏 elements
∧ and
∨ or
∀ for all
Notations

4. ⊂ subset
⊃ supersubset
∩ intersection
∪ union
𝑈 universal
∴ therefore
∵ because
~ equivalent
Notations

5. Try these:
Identify the elements of the given set
𝐴 = −9, −
48
12
, −
1
3
, − 5, 0,
5
6
,
12
4
, 18, 56
a. Natural numbers
b. Whole numbers
c. Negative integers
d. Integers
e. Rational numbers
f. Irrational numbers

6. Subtraction of real numbers is
defined in terms of addition as
follows:
If 𝑎 and 𝑏 are real numbers, the
difference of 𝑎 and 𝑏, denoted by
𝑎 − 𝑏 is a real number 𝑑 and 𝑎 −
𝑏 = 𝑑 if and only if 𝑎 = 𝑏 + 𝑑
Math Ideas to Remember

7. The real number system consists of
the set of real numbers and two
operation called addition and
multiplication.
Addition is denoted by the symbol +
while multiplication is denoted by ∙
or ×.
Math Ideas to Remember

8. Division of real numbers defined as
follows:
𝑎 and 𝑏 are real numbers, 𝑏 ≠ 0,
the quotient of 𝑎 and 𝑏 denoted by
𝑎 ÷ 𝑏 or
𝑎
𝑏
is a real number 𝑐 and
𝑎 ÷ 𝑏 = 𝑐 if and only if 𝑎 = 𝑏𝑐
Math Ideas to Remember

9. Basic Properties of Integers

10. Closure
𝑎 + 𝑏 and 𝑎 − 𝑏 are
integers whenever
𝑎 and 𝑏 are integers

11. Commutative Law
𝑎 + 𝑏 = 𝑎 + 𝑏 and
𝑎 ∙ 𝑏 = 𝑎 ∙ 𝑏 ∀ integers
𝑎 and 𝑏

12. Associative Law
𝑎 + 𝑏 + 𝑐 = 𝑎 + 𝑏 + 𝑐
and 𝑎 ∙ 𝑏 𝑐 = 𝑎 𝑏 ∙ 𝑐
∀ integers

13. Distributive Law
𝑎 + 𝑏 + 𝑐 = 𝑎 ∙ 𝑐 + 𝑏. 𝑐
∀ integers 𝑎, 𝑏 and 𝑐

14. Identity Elements
𝑎 + 0 = 𝑎 and 𝑎 ∙ 1 = 𝑎 ∀
integers 𝑎

15. Additive Inverse
For every integers 𝑎, there is an
integer solution 𝑥 to the equation at
𝑥 = 0. This integer 𝑥 is called the
additive inverse of 𝑎 and is denoted by
𝑎. By 𝑏 − 𝑎, we mean 𝑏 + (−𝑎).

16. Cancellation Law
If 𝑎, 𝑏 and 𝑐 are
integers with 𝑎 ∙ 𝑐 = 𝑏 ∙
𝑐 with 𝑐 ≠ 0 then 𝑎 = 𝑏.

17. Use the axioms for the integers to
prove the following statements for
all integers 𝑎, 𝑏 and 𝑐.
1. 𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎
2. (𝑎 + 𝑏)2
= 𝑎2
+ 2𝑎𝑏 + 𝑏2
3. 𝑎 + 𝑏 + 𝑐 = (𝑐 + 𝑎) + 𝑏
Exercises

18. Thank you for Listening…
“Without mathematics, there's nothing you can do. Everything around you
is mathematics. Everything around you is numbers.”
-Shakuntala Devi