 # Sets and Expressions Number Sets

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Sets and Expressions Number Sets
Natural (Counting) numbers – {1,2,3,…} Whole numbers – Natural numbers and 0, that is {0,1,2,3,….} Integers – Positive or negative whole numbers Rational numbers – any number that can be expressed as a/b, where a and b are integers Irrational numbers – any number that cannot be expressed as a fraction of two integers Real Numbers – The set of all numbers in each of the previous sets

The symbols is read “is an element of”, it is used to denote an element in a set is read “ is not an element of”, it is used to denote an element that is not part of the set

Set Builder Notation A way of writing a set according to conditions, the general form of which is:  𝑥 𝑥 ∈𝑠𝑒𝑡, 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 Translated as “x such that x is an element of a set, given some condition. Empty Set: also called a null set, a set that contains no elements. The empty set is represented by { } or Ø Do not use {Ø}, this means a set containing the element Ø Write the following in set builder notation: {…,0, 1, 2}

Roster Notation: A manner of writing a set in which all elements are listed, such as: {4, 5,6, …} Write the following in roster notation: 𝑥 𝑥 ∈𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠, 𝑥≤5 𝑥 𝑥 ∈𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠, 𝑥 ≥−4

Given the set {25, 7\3, -15, -3\4, √5, -3.7, 8.8,-99}
List the numbers in the set that belong to the set of Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers

Algebraic Expressions
An algebraic expression is any statement containing numbers and letters connected by an operator. Variable – A letter used to represent an unknown number Term – A number or product of a number and a variable Coefficient – A numerical factor of a term Constant – A term with no variables Example - Complete the table below Expression Variables Terms Coefficients Constants a. 3x – 5y + 3 b. -4x2 + 5y - 10 c. 7ab – c

Evaluating Algebraic Expressions
To evaluate an algebraic expression, substitute known values for variables into an expression and use the order of operations to simplify.

Order of Operations P- Parentheses or other grouping symbols
E- Exponents M- Multiplication D- Division A- Addition S- Subtraction From Left to Right From Left to Right

Examples:Evaluate the following expressions given the values for the variables:
Evaluate 2a + 3b when a = 2 and b = 7 Evaluate x2 + y2 – xy when x = -3 and y = 4 Evaluate when x = 9 and y = -2

Properties of Real Numbers:
Commutative Properties Addition: a+b = b+a Multiplication: a∙b = b∙a Associative Properties Addition: a+(b+c) = (a+b)+c Multiplication: (a∙b)∙c = a∙(b∙c) Addition and Multiplication Identities Additive Identity: a + 0 = 0 + a = a Multiplicative Identity: a∙1 = 1∙a = a Addition and Multiplication Inverses Additive Inverse: a + (-a) = 0 Multiplicative Inverse: a ∙ =1 Distributive Property of Multiplication a(b+c) = a∙b + b∙c

Examples: Give an example of each of the following properties:
a. Commutative __________________________ b. Associative __________________________ c. Distributive __________________________ d. Additive Identity ________________________ e. Multiplicative Identity_____________________ f. Additive inverse _____________________ g. Multiplicative Inverse _____________________

Examples: Name the identity illustrated:
b. -4 (6x) = (-4∙6 )x _____________________ c. y + (3+2) = (y+3) + 2 _____________________ d. 5∙ =1 _____________________ Use the distributive property to rewrite: a. 3(2x + 4) b. -2(3x + y – z) c. 5(2 + 3a – 4b)

Simplifying Algebraic Expressions
Algebraic expressions can be simplified by combining like terms. Like terms are two or more terms that have: a. The exact same variables b. Variables are raised to the same power Two or more like terms can be added by adding their coefficients. Examples – Simplify each algebraic expression:  a. 3y + 8y – 7 + 2    b. -7a + 3b – a – 17b – 20 c. 5(3x + 7) + 4x + 10     d. - (3a – 7) – 2(a + 8)