STRUCTURE: AXIOMS
Example 2: Associative Property
10(5 ∙ 2 ) = (10 ∙ 5)2 10(x ∙ 3) = (10 ∙ x)3
Example 1: Commutative Property
2 + 3 = 3 + 2 2 + x = x + 2 x + (3 + y) = (3 + y) + x
Example 4: Additive Inverse Property
3 + (-3) = 0 x + (-x) = 0
Example 2: Associative Property
3 + (5 + 2) = (3 + 5) + 2 4 + (2 + x) = (4 + 2) + x (x + 4) + (y + 3) = (x + 4 + y) + 3
Example 3: Identity Property
3 ∙ 1 = 3 x ∙ 1 = x
Example 5: Distributive Property
4(8 + 2) = (4 ∙ 8) + (4 ∙ 2) 3(5 - 2) = (3 ∙ 5) - (3 ∙ 2) a(x + 2) = (a ∙ x) + (2 ∙ a ) or ax + 2a
1. 8 = 8 2. If x = y, then y = x 3. If x = y and y = 7, then x = 7 4. 6 + 2 = 2 + 6 5. 5 ∙ 2 = 2 ∙ 5 6. 10 + (3 + 2) = (10 + 3) + 2 7. 6(3 ∙ 5) = (6 ∙ 3)5 8. 4(8 + 1) = 4 ∙ 8 + 4 ∙ 1 9. 6 + 0 = 6 10. 8 ∙ 1 = 8 11. 2 = 1 12. 5 ∙ 0 = 0
4. commutative - addition 9. identity - addition 8. distributive 7. associative - multiplication 11. multiplicative inverse 2. symmetric 5. commutative - multiplication 1. reflexive 12. property of zero 3. transitive 6. associative - addition 10. identity - multiplication
Example 1: Commutative Property
5 ∙ 7 = 7 ∙ 5 x ∙ 2 = 2 ∙ x 5(x + y) = (x + y)5
Example 4: Multiplicative Inverse Property
5 ∙ = 1 AB ∙ = 1; AB 0 x ∙ = 1; x 0
Example 3: Identity Property
6 + 0 = 6 x + 0 = x (a + b) + 0 = a + b
Example 6: Zero Property
6 ∙ 0 = 0 x ∙ 0 = 0 (a - b) 0 = 0
PROPERTIES OF ADDITION
A. commutative a + b = b + a B. associative a + (b + c) = (a + b) + c C. identity a + 0 = a D. additive inverse a + (-a) = 0
PROPERTIES OF MULTIPLICATION
A. commutative a ∙ b = b ∙ a B. associative a(b ∙ c) = (a ∙ b)c C. identity a ∙ 1 = a D. multiplicative inverse a ∙ = 1; a 0 E. distributive a(b + c) = (a ∙ b) + (a ∙ c) F. zero a ∙ 0 = 0
GENERAL PROPERTIES
A. reflexive property a = a B. symmetric property if a = b, then b = a C. transitive property if a = b and b = c, then a = c
More Properties
If a, b, c R where R is a set of real numbers, then the following statements are true.
THE CLOSURE AXIOM
Notice that if we add or multiply two whole numbers, the result is another whole number. However, if we subtract two whole numbers, the result is not necessarily another whole number: 5 - 7 = -2, an Integer. Similarly, if we divide two whole numbers, we do not always get another whole number: 3/2 = 1.5, not a whole number! Given a set and an operation, we say that the set has closure for that operation if we can guarantee that the result of performing that operation on any elements in the set will result in another element of that same set. For the Whole numbers, addition and multiplication are closed. Subtraction and division are not. For the Integers, addition, multiplication and subtraction are closed. Division is not. For the Real numbers, addition, multiplication, subtraction and division are all closed.
Theorems
Theorems are statements about mathematics requiring proof. Proof starts from the axioms and follows a logical path to the theorem you want to prove.
Choose 2 axioms that allows 22 + (m + 8) to be written as m + 30
commutative - addition distributive
Choose the axiom that allows b(7) to be written 7b.
distributive
Select the property that allows the statement 3 = x to be written x = 3.
symmetric