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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 Page 2 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 Page 3 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 Page 4