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Questions and Answers
In mathematical language, which of the following best represents a complete thought?
In mathematical language, which of the following best represents a complete thought?
- Expression
- Noun
- Equation (correct)
- Operation
When translating 'Ten less than a number' into a mathematical expression, which of the following is correct?
When translating 'Ten less than a number' into a mathematical expression, which of the following is correct?
- $10 + x$
- $10 - x$
- $x - 10$ (correct)
- $x + 10$
Which of the following uses of numbers is considered 'nominal'?
Which of the following uses of numbers is considered 'nominal'?
- Counting the number of students in a class.
- Measuring the height of a building.
- Using a jersey number to identify a player. (correct)
- Determining the position of a runner in a race.
In the context of mathematical language, what does the word 'is' most commonly represent?
In the context of mathematical language, what does the word 'is' most commonly represent?
Which of the following mathematical symbols is also a letter in the English alphabet?
Which of the following mathematical symbols is also a letter in the English alphabet?
If 'and' is used to mean addition, how would you translate 'five and y increased by two' into a mathematical expression?
If 'and' is used to mean addition, how would you translate 'five and y increased by two' into a mathematical expression?
Which of the following is an example of an ordinal number?
Which of the following is an example of an ordinal number?
What is the primary function of 'grouping symbols' in a mathematical expression?
What is the primary function of 'grouping symbols' in a mathematical expression?
In a coordinate plane, which quadrant is defined by negative x-values and positive y-values?
In a coordinate plane, which quadrant is defined by negative x-values and positive y-values?
Given a relation represented as a set of ordered pairs, what aspect of the ordered pairs determines whether the relation is a function?
Given a relation represented as a set of ordered pairs, what aspect of the ordered pairs determines whether the relation is a function?
Which of the following statements accurately describes the relationship between a relation and a function?
Which of the following statements accurately describes the relationship between a relation and a function?
Given the relation {(-2, 3), (-1, 4), (0, 5), (1, 6), (2, 7)}, what is the range of this relation?
Given the relation {(-2, 3), (-1, 4), (0, 5), (1, 6), (2, 7)}, what is the range of this relation?
Consider the mapping of inputs to outputs. If an input value in a relation has multiple output values associated with it, what can be concluded?
Consider the mapping of inputs to outputs. If an input value in a relation has multiple output values associated with it, what can be concluded?
Which of the sets of ordered pairs below represents a function?
Which of the sets of ordered pairs below represents a function?
In the equation $y = 2x$, if $x$ is the input and $y$ is the output, what is the output when the input is $-3$?
In the equation $y = 2x$, if $x$ is the input and $y$ is the output, what is the output when the input is $-3$?
Given the following data: Input X: 1, 2, 1 and Output Y: 6, 7, 7. Does this data represent a function? Choose the most accurate answer.
Given the following data: Input X: 1, 2, 1 and Output Y: 6, 7, 7. Does this data represent a function? Choose the most accurate answer.
Which of the following sets is defined using set builder notation?
Which of the following sets is defined using set builder notation?
Which of the following statements accurately describes the difference between equivalent and equal sets?
Which of the following statements accurately describes the difference between equivalent and equal sets?
Given a universal set U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and a set A = {2, 4, 6, 8}, what is the complement of A (A')?
Given a universal set U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and a set A = {2, 4, 6, 8}, what is the complement of A (A')?
If set A = {1, 3, 5} and set B = {1, 2, 3, 4, 5, 6}, which of the following statements is true?
If set A = {1, 3, 5} and set B = {1, 2, 3, 4, 5, 6}, which of the following statements is true?
What does the intersection of two sets, A and B, represent?
What does the intersection of two sets, A and B, represent?
Which of the following is an example of an infinite set?
Which of the following is an example of an infinite set?
Given set A = {2, 4, 6} and set B = {4, 6, 8}, what is the union of A and B (A∪B)?
Given set A = {2, 4, 6} and set B = {4, 6, 8}, what is the union of A and B (A∪B)?
In the context of relations, what does the 'domain' refer to?
In the context of relations, what does the 'domain' refer to?
What is a relation between two variables x and y defined as?
What is a relation between two variables x and y defined as?
Which of the following correctly represents the set of rational numbers (Q) using set-builder notation?
Which of the following correctly represents the set of rational numbers (Q) using set-builder notation?
In the equation $y = 2x$, representing the number of shoes ($y$) in $x$ pairs, how does the domain relate to the scenario?
In the equation $y = 2x$, representing the number of shoes ($y$) in $x$ pairs, how does the domain relate to the scenario?
If Mr. Landry drives at an average speed of 55 mph, represented by the function $d = 55t$, what does the range signify in this context?
If Mr. Landry drives at an average speed of 55 mph, represented by the function $d = 55t$, what does the range signify in this context?
At Pete’s Pizza Parlor, the cost of a pizza is determined by the number of toppings added. If the base price is $5 and each topping costs $1.50, what does the dependent variable represent?
At Pete’s Pizza Parlor, the cost of a pizza is determined by the number of toppings added. If the base price is $5 and each topping costs $1.50, what does the dependent variable represent?
Which of the following best describes the role of logic in mathematics?
Which of the following best describes the role of logic in mathematics?
Which statement accurately describes a proposition in the context of elementary logic?
Which statement accurately describes a proposition in the context of elementary logic?
Consider the statement: 'If it is raining, then the ground is wet.' Which logical connective does this statement exemplify?
Consider the statement: 'If it is raining, then the ground is wet.' Which logical connective does this statement exemplify?
If $p$ represents the proposition 'The car is red,' what does $\neg p$ (not p) represent?
If $p$ represents the proposition 'The car is red,' what does $\neg p$ (not p) represent?
Given two propositions, P: 'The sun is shining' and Q: 'Birds are singing', which logical connective would you use to symbolize 'The sun is shining and birds are singing'?
Given two propositions, P: 'The sun is shining' and Q: 'Birds are singing', which logical connective would you use to symbolize 'The sun is shining and birds are singing'?
Which of the following statements correctly describes the condition for a conjunction ($p \land q$) to be true?
Which of the following statements correctly describes the condition for a conjunction ($p \land q$) to be true?
If $p$ is '5 is an even number' and $q$ is '6 is divisible by 3', what is the truth value of the proposition $p \land q$?
If $p$ is '5 is an even number' and $q$ is '6 is divisible by 3', what is the truth value of the proposition $p \land q$?
In what scenario is the disjunction of two propositions ($p \lor q$) false?
In what scenario is the disjunction of two propositions ($p \lor q$) false?
Given $p$: 'The sky is green' and $q$: 'Birds can fly', determine the truth value of $p \lor q$.
Given $p$: 'The sky is green' and $q$: 'Birds can fly', determine the truth value of $p \lor q$.
What is the negation of the statement 'All cats are black'?
What is the negation of the statement 'All cats are black'?
If $p$ is 'Today is Monday', which of the following correctly expresses $\neg p$?
If $p$ is 'Today is Monday', which of the following correctly expresses $\neg p$?
Consider the statement: 'Students who have taken calculus can enroll in advanced physics'. If $p$ represents 'students have taken calculus' and $q$ represents 'students can enroll in advanced physics', how would you symbolically represent this statement?
Consider the statement: 'Students who have taken calculus can enroll in advanced physics'. If $p$ represents 'students have taken calculus' and $q$ represents 'students can enroll in advanced physics', how would you symbolically represent this statement?
Given $p$: 'The number 7 is even' and $q$: 'The number 8 is even', evaluate the truth value of $\neg p \land q$.
Given $p$: 'The number 7 is even' and $q$: 'The number 8 is even', evaluate the truth value of $\neg p \land q$.
Given the statement 'If it is raining, then the ground is wet,' which of the following represents its contrapositive?
Given the statement 'If it is raining, then the ground is wet,' which of the following represents its contrapositive?
Which of the following statements is logically equivalent to 'p only if q'?
Which of the following statements is logically equivalent to 'p only if q'?
Consider the implication: 'If a number is divisible by 4, then it is even.' Which type of implication does this represent?
Consider the implication: 'If a number is divisible by 4, then it is even.' Which type of implication does this represent?
Which of the following best describes a 'causal implication'?
Which of the following best describes a 'causal implication'?
Given that p is false and q is true, what is the truth value of the implication $p \rightarrow q$?
Given that p is false and q is true, what is the truth value of the implication $p \rightarrow q$?
Which of the following represents the inverse of the statement 'If it is sunny, then I will go for a walk'?
Which of the following represents the inverse of the statement 'If it is sunny, then I will go for a walk'?
Which of the following conditions is sufficient for q?
Which of the following conditions is sufficient for q?
Consider the statement: 'Students who have taken Algebra can take Discrete Mathematics.' According to the content, if a certain student has not taken Algebra but is taking Computer Science, and can take Discrete Mathematics, what are the truth values of p and q respectively?
Consider the statement: 'Students who have taken Algebra can take Discrete Mathematics.' According to the content, if a certain student has not taken Algebra but is taking Computer Science, and can take Discrete Mathematics, what are the truth values of p and q respectively?
Flashcards
Language
Language
A system of symbols (spoken, signed, or written) used for communication within a social group.
Mathematical Symbols
Mathematical Symbols
The set of symbols (English alphabet, numerals, Greek letters, etc.) used to express mathematical ideas.
Symbols: English vs. Math
Symbols: English vs. Math
English uses alphabet and punctuation while mathematics includes numerals, Greek letters and special symbols.
Name
Name
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Complete Thought
Complete Thought
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Action
Action
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Math Language Difficulties
Math Language Difficulties
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Cardinal Numbers
Cardinal Numbers
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What is a Set?
What is a Set?
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Explicitly Specifying a Set
Explicitly Specifying a Set
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Implicitly Specifying a Set
Implicitly Specifying a Set
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Set Builder Notation
Set Builder Notation
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Infinite Sets
Infinite Sets
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Equivalent Sets
Equivalent Sets
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Equal Sets
Equal Sets
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Intersection of Sets
Intersection of Sets
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Union of Sets
Union of Sets
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Complement of a Set
Complement of a Set
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Domain
Domain
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Range
Range
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Independent Variable
Independent Variable
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Dependent Variable
Dependent Variable
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Proposition
Proposition
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Simple Proposition
Simple Proposition
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Compound Proposition
Compound Proposition
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Negation
Negation
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Range (in functions)
Range (in functions)
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Coordinate Plane
Coordinate Plane
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Graph (Ordered Pairs)
Graph (Ordered Pairs)
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Function
Function
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Function Test (x-coordinates)
Function Test (x-coordinates)
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Mapping (Functions)
Mapping (Functions)
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Negation (¬p)
Negation (¬p)
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Conjunction (p ∧ q)
Conjunction (p ∧ q)
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Disjunction (p ∨ q)
Disjunction (p ∨ q)
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Conjunction Example
Conjunction Example
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Disjunction Example
Disjunction Example
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Negation Example
Negation Example
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Disjunction Truth Value
Disjunction Truth Value
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p ∨ q
p ∨ q
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Implication (p → q)
Implication (p → q)
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Hypothesis (Antecedent)
Hypothesis (Antecedent)
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Conclusion (Consequence)
Conclusion (Consequence)
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Causal Implication
Causal Implication
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Definitional Implication
Definitional Implication
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Converse
Converse
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Inverse
Inverse
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Study Notes
- This lecture explores the nature of mathematics, focusing on mathematical language, symbols, sets, and elementary logic
Language of Mathematics
- Language is a system of conventional symbols used for communication, be they spoken, signed or written
- Symbols in mathematics include:
- "+" for plus
- "-" for minus
- "x" or "*" for multiplied by
- "÷" for divided by
- "+/-" for plus or minus
- "<" or ">" is greater or less than
- "i" for imaginary
- "=" for equal to
- "≠" for not equal to
- "=>" implies
- "≡" is equivalent to
- "∞" for infinity
- "â–³" triangle or delta, etc
- In English language, symbols include the English alphabet and punctuation
- In mathematics, symbols include English and Greek alphabets, numerals, grouping symbols, and special symbols
- In English, there are nouns and sentences
- In mathematics, there are expressions and equations
- English language has verbs
- Mathematical language has equality, inequality and membership in a set
- Difficulties in math language:
- Words can have different meanings in math compared to English
- "and" means plus
- "is" can have different meanings
- Numbers can be cardinal (counting), ordinal (positions), or nominal (identification)
Keywords in Math Problems
- Addition (+): increased by, more than, combined, together, total of, sum, plus and added to, comparatives like "greater than"
- Subtraction (-): decreased by, minus, less, difference between/of, less than, fewer than, left, left over, after, save and comparatives like "smaller than"
- Multiplication (x): of, times, multiplied by, product of, increased/decreased by a factor of, twice, triple, each
- Division (÷): per, a, out of, ratio of, quotient of, percent (divide by 100), equal pieces, split, average
- Equals (=): is, are, was, were, will be, gives, yields, sold for, cost
- Keywords for inequalities include minimum, maximum, greater than or less than, that help determine if and when something is more or less than something else
Translating Between English and Math
- "Four more x" translates to 4 + x
- "Six more than a number" converts to x + 6
- "Seven greater than a number" is x + 7
- "Two subtracted from a number" is expressed as x - 2
- "Eight times a number is forty-eight" becomes 8x = 48
Sets
- A set in math is a collection of well-defined and distinct objects, considered as an object in its own right
- Objects of a set are listed or described, separated by commas, and enclosed in braces
- Sets are represented using capital letters
- Objects in a set are elements or members
- Sets can be specified by:
- Listing elements explicitly, such as C = { a, o, i }
- Listing elements implicitly, such as K = { 10, 15, 20, 25, ..., 95 }
- Using set builder notation, like Q = { x | x = p/q where p and q are integers and q ≠0 }
- Membership Relation:
- x ∈ A means x is a member of set A
- x ∉ A means x is not a member of set A
Types of Sets
- Infinite sets are conceptually not finite
- Special infinite sets include:
- Natural numbers (N) = {0, 1, 2, ...}
- Integers (Z) = {..., -2, -1, 0, 1, 2, ...}
- Real numbers (R), like 374.1828471929498181917281943125...
- Cardinality indicates the number of elements in a set
- If A = {5}, |A| = 1
- Equivalent sets have the same cardinality
- Equal sets have identical elements
- A = {4, 5, 6} and C = {6, 5, 4} are equal
- Using Venn diagrams can illustrate the relationships between sets
- Key set operations:
- Intersection (A ∩ B): Members common to both sets
- Union (A ∪ B): Members in either set or both
- Complement (A'): Members not in the set
Relations and Functions
- A relation is a set of ordered pairs (x, y) x is an independent variable, known as "domain" y is a dependent variable, known as "range"
- Relations can be represented in ordered pairs, equations or sentences
- A function dictates every input (x) has exactly one output (y)
- Focus on x to determine function
- y has no bearings on determining functions
- Vertical Line Test. This test can determine if a graph is a Function
- If a vertical line intersects the graph at only one point, it is a Function
Concepts
- Proposition is a statement that is either true or false
- Simple propositions convey a single idea
- Compound propositions convey two or more ideas
- Logical connectives determine how statements relate
Truth Tables
- Negation of P:
- If P is true, ¬P is false, and vice versa
- Conjunction of P and Q (P ∧ Q):
- Only true if both P and Q are true
- Disjunction of P and Q (P ∨ Q):
- True if either P or Q or both are true
- Implications
- If P, then Q
- P implies Q
- P→Q (If P, then Q) is only false when P is true and Q is false; otherwise, it is true
Associated Conditional Statements
- Converse: Switch hypothesis and conclusion
- Inverse: Negate both hypothesis and conclusion
- Contrapositive: Negate and switch hypothesis and conclusion
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