Questions and Answers
What are natural numbers?
Whole numbers include which of the following?
Integers are defined as:
Sets are composed of what?
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What does the symbol ∈ signify?
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What characterizes a finite set?
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What is an infinite set?
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Sets can be designated in one of two ways, which are:
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What are subsets?
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What is the union of two sets?
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How is the intersection of two sets defined?
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What does D C imply in terms of set operations?
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What are theorems in mathematics?
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Which of the following are examples of general properties?
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Which properties of addition are correctly matched?
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Identify the properties of multiplication.
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What does PEMDAS stand for?
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What is the domain in a relation or function?
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Define a function.
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What is the input in a relation or function?
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What does output represent in a function?
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What is the range in terms of relations or functions?
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What defines a relation?
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How can the domain and range be listed?
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What does a graph of an ordered-pair number represent?
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What is an exponent?
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Which of the following is the correct exponent theorem?
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Study Notes
Number Types and Sets
- Natural Numbers: Counting numbers starting from one and extending to infinity.
- Whole Numbers: Includes all natural numbers plus zero, ranging from zero to infinity.
- Integers: Comprise all positive and negative whole numbers including zero, spanning from negative infinity to positive infinity.
- Sets: Collections of distinct objects called members or elements.
Set Notation and Types
- Symbol ∈: Indicates membership, read as "is a member of."
- Finite Set: Contains a specific number of members.
- Infinite Set: Has an unlimited number of members.
- Methods of Designating Sets: Utilizes either the list method, which explicitly lists elements, or the rule method using set-builder notation.
- Subsets: Sets that are contained within other sets.
Set Operations
- Union of Two Sets (A ∪ B): A set that contains elements from either Set A or Set B without duplication.
- Intersection of Two Sets (A ∩ B): A set comprising elements that are common to both Set A and Set B.
- If D ⊆ C (D is a subset of C): The intersection of C and D equals D (C ∩ D = D).
Mathematical Theorems and Properties
- Theorems: Statements within mathematics that require proof, originating from axioms to logical conclusions.
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General Properties:
- Reflexive Property: ( a = a ).
- Symmetric Property: If ( a = b ), then ( b = a ).
- Transitive Property: If ( a = b ) and ( b = c ), then ( a = c ).
Arithmetic Properties
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Properties of Addition:
- Commutative: ( a + b = b + a ).
- Associative: ( a + (b + c) = (a + b) + c ).
- Identity: ( a + 0 = a ).
- Additive Inverse: ( a + (-a) = 0 ).
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Properties of Multiplication:
- Commutative: ( a \cdot b = b \cdot a ).
- Associative: ( a(b \cdot c) = (a \cdot b)c ).
- Identity: ( a \cdot 1 = a ).
- Multiplicative Inverse: ( a \cdot \frac{1}{a} = 1 ) for ( a \neq 0 ).
- Distributive: ( a(b + c) = (a \cdot b) + (a \cdot c) ).
- Zero Property: ( a \cdot 0 = 0 ).
Function Basics
- Order of Operations: Follow the PEMDAS hierarchy: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
- Domain: Refers to the first element (x-value) in a relation or function, also known as the input.
- Function: A relation where each x-value corresponds to one unique y-value; no x-value is repeated.
- Input: The x-value of a relation or function.
- Output: The y-value corresponding to the input in a relation or function.
- Range: The set of all second elements (y-values) of a relation or function, also known as the output.
Relations
- Relation: A collection of numbers that can be graphed on a coordinate plane; can be a function but is not required to be.
- Domain and Range Representation: Can be listed based on the given order or rearranged numerically; no need to list duplicates.
- Graphing Ordered Pairs: Each ordered pair represents a point on the coordinate axes, with the first element indicating the x-axis and the second the y-axis.
Exponents
- Exponent: A small number positioned above and to the right of a base, indicating how many times the base is a factor.
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Exponent Theorems:
- When multiplying: ( a^m \cdot a^n = a^{m+n} ).
- When dividing: ( \frac{a^m}{a^n} = a^{m-n} ).
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Test your understanding of the foundational elements of Algebra 2 through these flashcards. Explore key concepts including natural numbers, whole numbers, and integers. Perfect for reviewing essential definitions and preparing for exams.
