Download Chapter 1: Sets, Operations and Algebraic Language

Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
1) Commutative Properties of Addition and Multiplication:
 The order in which you add or multiply does not matter.
a + b = b + a and
a∙b=b∙a
 Examples:
2) Symmetric Property:
 If a + b = c, then c = a + b
3) Reflexive Property:
 a+b=a+b
 Nothing changes
4) Associative Properties of Addition and Multiplication.
 The grouping of addition and multiplication does not matter. (Parenthesis)
 Examples:
5) Transitive Property:
 If a = b and b = c, then a = c.
 If, and, then
 Examples:
6) Distributive Property:
 a (b + c) = ab + ac
 Examples:
and
a(b – c) = ab – ac
Combining Like Terms
Like terms are terms such as 4x and 5x that differ only in their numerical coefficients.
To combine 4x and 5x, use the reverse of the distributive property:
 4 x  5 x  x(4  5)  9 x
To add or subtract like terms, simply combine their numerical coefficients:
9y  7 y 
7b  b 
2 xy  3xy  4 xy 
Chapter 1, Section 1.5
Page 1
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
7) Additive Identity:
 When zero is added to any number or variable, the sum is the number
or variable.
 a+0=a
8) Multiplicative Identity:
 When any number or variable is multiplied by 1, the product is the
number or variable.
 a∙1=a
9) Multiplicative Property of Zero:
 When any number or variable is multiplied by zero, the product is
0.
 a∙0=0
10) Closure Property
Closure is a property (or characteristic) of a set of numbers, such as integers, and an
operation, such as addition, subtraction, multiplication or division.
Closed
Same Output Set
Given
Operation
e
Same S
Given Input Set
A set and an operation are
closed if the inputs and the
outputs of the operation are
from the same set.
Example: The set of
integers is closed to the
operation of addition.
t
Not Closed
Different Output Set
Given
Operation
nt
Differe
Chapter 1, Section 1.5
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Sets
Given Input Set
A set and an operation are
not closed if the inputs and
the outputs of the operation
are from the different sets.
Example: The set of
integers is not closed to the
operation of division.
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
Examples
Which expression is an example of the associative property?
(1) (x + y) + z = x + (y + z)
(2) x + y + z = z + y + x
(3) x(y + z) = xy + xz
(4) x • 1 = x
Solution:
The operation * for the set {p,r,s,v} is defined in the accompanying table. What is the
inverse element of r under the operation *?
(1) p
(2) r
Solution
(3) s
(4) v
Which set is closed under division?
(1) {1}
(3) integers
(2) counting numbers
(4) whole numbers
Solution:
Chapter 1, Section 1.5
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Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
Do Now Questions
1 Which expression is an example of the associative property?
(1) (x + y) + z = x + (y + z)
(2) x + y + z = z + y + x
(3) x(y + z) = xy + xz
(4) x • 1 = x
2 Which equation illustrates the associative property of addition?
(1) x + y = y + x
(2) 3(x + 2) = 3x + 6
(3) (3 + x) + y = 3 + (x + y)
(4) 3 + x = 0
3 Which equation illustrates the associative property?
(1) a (1)  a
(2) a  b  b  a
(3) a (b  c)  (ab)  (ac)
(4) (a  b)  c  a  (b  c)
4
Which equation illustrates the distributive property of multiplication over
addition?
(1) 6(3a + 4b) = 18a + 4b
(2) 6(3a + 4b) = 18a + 24b
(3) 6(3a + 4b) = (3a + 4b)6
(4) 6(3a + 4b) = 6(4b + 3a)
5
Which equation illustrates the distributive property?
(1) 5(a + b) = 5a + 5b
(2) a + b = b + a
(3) a + (b + c) = (a + b) + c
(4) a + 0 = a
Chapter 1, Section 1.5
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