Algebra I - Sets and Functions Overview
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Questions and Answers

What does g(b) equal when the preimage f^{-1}({b}) is empty?

  • a, the unique element in the preimage of b
  • undefined
  • b, the element in the codomain
  • α, a defined constant (correct)

What does it mean for set A to be a proper subset of set B?

  • Every element of A is also in B and A contains all elements of B.
  • Every element of A is also in B, but there is at least one element in B that is not in A. (correct)
  • There exists an element in A that is not in B.
  • A and B contain exactly the same elements.

How does the injective property of f affect the preimage f^{-1}({b})?

  • It can contain multiple elements.
  • It contains exactly one element. (correct)
  • It cannot contain any elements.
  • It contains at least one element.

Which statement correctly describes the relationship between the empty set and any set A?

<p>The empty set is a subset of every set. (C)</p> Signup and view all the answers

Which of the following statements is true regarding functions with a right-inverse?

<p>A function is surjective if it has a right-inverse. (B)</p> Signup and view all the answers

What can be concluded if a function f has both a left inverse and a right inverse?

<p>The function f is bijective. (B)</p> Signup and view all the answers

How can it be determined that two sets A and B are equal?

<p>If every element of A is in B and every element of B is in A. (B)</p> Signup and view all the answers

If g is a left-inverse of f, which statement must hold true?

<p>g(f(a)) = a for all a ∈ A. (D)</p> Signup and view all the answers

What can be inferred from the statement A ⊆ B?

<p>All elements of A are also in B. (D)</p> Signup and view all the answers

What is true about the relationship between one-sided inverses of a function?

<p>They are identical if both exist. (B)</p> Signup and view all the answers

If set A contains the elements {1, 2} and B contains the elements {1, 2, 3}, what can be said about A and B?

<p>A is a proper subset of B. (C)</p> Signup and view all the answers

What is the power set of a set A?

<p>The set of all possible subsets of A. (C)</p> Signup and view all the answers

Which condition guarantees that f has a right-inverse?

<p>f is surjective. (A)</p> Signup and view all the answers

Which of the following statements is true regarding power sets?

<p>If A ⊆ B, then P(A) ⊆ P(B). (C)</p> Signup and view all the answers

In which case does the preimage f^{-1}(b) contain no elements?

<p>b is not in the image of f. (A)</p> Signup and view all the answers

What condition must be satisfied for A to be a subset of B?

<p>Every element in A must also be an element of B. (B)</p> Signup and view all the answers

If both functions f and g are surjective, what can be concluded about the composition g ◦ f?

<p>g ◦ f is surjective (A)</p> Signup and view all the answers

What is true about the composition of two bijective functions f and g?

<p>g ◦ f is bijective (C)</p> Signup and view all the answers

If g ◦ f is surjective, what can be inferred about function g?

<p>g must be surjective (D)</p> Signup and view all the answers

What must be true for a function f to have a left-inverse?

<p>f must be injective (B)</p> Signup and view all the answers

If f(a1) = f(a2) for a1, a2 in set A, and f has a left-inverse, what can be concluded?

<p>a1 and a2 must be equal (C)</p> Signup and view all the answers

Which statement is true concerning the identity function iA?

<p>iA(a) = a for all a ∈ A (B)</p> Signup and view all the answers

What is the necessary condition for g to be a right-inverse of f?

<p>f ◦ g must equal the identity function iB (D)</p> Signup and view all the answers

If g is injective and g(f(a1)) = g(f(a2)), what can be concluded about a1 and a2?

<p>a1 = a2 (D)</p> Signup and view all the answers

What condition is necessary for the function h to be injective?

<p>h(x) = h(y) implies x = y (C)</p> Signup and view all the answers

Which equation does not hold true given that ad ≠ bc?

<p>dy - b = 0 when y = ac (C)</p> Signup and view all the answers

What does the equation (cy - d)(ax - b) = (ay - b)(cx - d) represent?

<p>A necessary condition for injectivity (C)</p> Signup and view all the answers

What is implied by the statement 'x = cy - a'?

<p>It indicates a linear relationship between x and y (C)</p> Signup and view all the answers

Why must dy - b not equal 0 when evaluating x?

<p>It keeps the denominator non-zero (B)</p> Signup and view all the answers

In the context of the given information, what does it mean for y ∈ R , ac?

<p>y cannot equal ac (D)</p> Signup and view all the answers

What conclusion can be drawn from the equation (ad - bc)y = (ad - bc)x?

<p>It implies y = x when ad ≠ bc (C)</p> Signup and view all the answers

What is the significance of the condition ad - bc ≠ 0?

<p>It ensures that x and y can be uniquely determined (C)</p> Signup and view all the answers

What is the complement of a set A denoted as Ac?

<p>The set of elements in the universal set U that are not in A (C)</p> Signup and view all the answers

Which of the following statements about the Cartesian product A × B is true?

<p>It can produce ordered pairs by combining elements from A and B (A)</p> Signup and view all the answers

Which of the following correctly describes De Morgan's Laws?

<p>(A ∪ B)c = Ac ∩ Bc and (A ∩ B)c = Ac ∪ Bc (B)</p> Signup and view all the answers

If A and B are finite sets with |A| = 3 and |B| = 4, what is the size of the Cartesian product |A × B|?

<p>12 (D)</p> Signup and view all the answers

What is true about the intersection of a set A with its complement Ac?

<p>It is equal to the empty set (B)</p> Signup and view all the answers

Given the sets A = {1, 2} and B = {3, 4}, what is A × B?

<p>{(1, 3), (1, 4), (2, 3), (2, 4)} (A)</p> Signup and view all the answers

What does the equation A ∪ Ac equal to?

<p>U (B)</p> Signup and view all the answers

What property does the Cartesian product satisfy?

<p>A × (B ∪ C) = A × B ∩ A × C (D)</p> Signup and view all the answers

What can be concluded if $f(a) eq b$ for some $b eq f(a)$?

<p>$f(a)$ is in $D1 ext{ or } D2$ (C)</p> Signup and view all the answers

What is the outcome of the composition $g ullet f$?

<p>$g ullet f$ is defined as $g(f(a))$ for all $a$ in $A$ (A)</p> Signup and view all the answers

Which of the following defines the restriction of a function?

<p>A function defined for a smaller subset of the original domain (A)</p> Signup and view all the answers

If $g$ is a restriction of $h$, what can be inferred about their outputs?

<p>$h$ can produce outputs not in the range of $g$ (C)</p> Signup and view all the answers

Why is $g ullet f$ confirmed to be a function from $A$ to $C$?

<p>Every input in $A$ maps to a unique output in $C$ (C)</p> Signup and view all the answers

Which of the following correctly represents the set of ordered pairs for the composition $g ullet f$?

<p>${(a, g(f(a))) ext{ for all } a ext{ in } A}$ (A)</p> Signup and view all the answers

If $a otin f^{-1}(D1 ext{ or } D2)$, what implication can be drawn?

<p>$f(a)$ cannot be in either $D1$ or $D2$ (C)</p> Signup and view all the answers

How is the intersection of the images of two sets $f(C1)$ and $f(C2)$ characterized?

<p>It represents shared outputs from these functions (C)</p> Signup and view all the answers

Flashcards

Complement of a set A

The set of all elements in the universal set U that are not in set A.

De Morgan's Laws for sets

Rules that relate the complements of unions and intersections of sets.

Cartesian product of sets A and B

A set containing all ordered pairs (a, b) where 'a' is from set A and 'b' is from set B.

Ordered pair

A list of two elements (x, y) in a specific order.

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Cartesian product properties

Rules for manipulating Cartesian products of sets.

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Set properties (subset)

Rules for relationships between sets like subset and union/intersection.

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Set properties (empty set)

How empty set interacts with union and intersection of sets.

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Set properties (universal set)

Describes the relationship between set A and the universal set.

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Subset (⊆)

Set A is a subset of set B if every element of A is also an element of B.

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Proper Subset (⊂)

Set A is a proper subset of set B if A is a subset of B, AND there's at least one element in B that is not in A.

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Equal Sets (A = B)

Two sets A and B are equal if and only if they have exactly the same elements.

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Power Set (P(A))

The power set of a set A is the set containing all possible subsets of A, including the empty set.

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Empty Set (∅)

The set containing no elements.

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A ⊆ B Implies P(A) ⊆ P(B)

If set A is a subset of set B, then the power set of A is a subset of the power set of B.

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∀x(x ∈ A ⇐⇒ x ∈ B)

For all x, x is in A if and only if x is in B.

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A ⊆ B

all elements of A are also elements of B.

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Composite Function (g ◦ f)

A function formed by applying one function (f) to an input, then applying another function (g) to the result.

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Injective g ◦ f

If two inputs to (g ◦ f) produce the same output, then those two inputs must be equal. (g ◦ f) maps distinct inputs to distinct outputs.

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Surjective g ◦ f

Every element in the codomain of (g ◦ f) is an output of (g ◦ f) for some input.

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Bijective g ◦ f

(g ◦ f) is both injective (one-to-one) and surjective (onto).

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Injective f if g ◦ f is injective

If the composite function (g ◦ f) is injective, then the inner function (f) must also be injective.

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Surjective g if g ◦ f is surjective

If the composite function (g ◦ f) is surjective, then the outer function (g) must also be surjective.

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Left-Inverse of f

A function (g) that 'undoes' f when applied to the left of f (g ◦ f = iA).

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Right-Inverse of f

A function (g) that 'undoes' f when applied to the right of f (f ◦ g = iB).

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Restriction of a Function

A function f|D, obtained from f: A->B by limiting its domain to a subset D ⊆ A. f|D contains only pairs (x, y) where x ∈ D and y = f(x).

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Extension of a Function

A function h is an extension of a function g if g is a restriction of h. Essentially, h has a larger domain than g, but they behave identically on g's domain.

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Composition of Functions

A new function formed by combining two functions f: A->B and g: B->C. g∘f operates by first applying f to an input x ∈ A, then applying g to the result f(x) ∈ B. The final result is in C.

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What does g∘f map?

g∘f maps elements from the domain of f to the codomain of g.

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How to verify g∘f is a function

To prove g∘f is a function, you must show that every input a ∈ A maps to exactly one output c ∈ C. This involves showing that if (a, c1) and (a, c2) are in g∘f, then c1 must equal c2.

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f (a) ∈ D1 ∪ D2

This means that f(a) is either in D1 or in D2, or possibly in both sets.

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a ∈ f −1 (D1 ) ∪ f −1 (D2 )

This means that a is either in the preimage of D1 or in the preimage of D2, or possibly in both sets.

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f −1 (D1 ∪ D2 ) = f −1 (D1 ) ∪ f −1 (D2 )

The preimage of the union of two sets is equal to the union of the preimages of those sets.

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Left-Inverse of Function

A function 'g' is a left-inverse of function 'f' if applying 'g' after 'f' returns the original input. This means g(f(a)) = a for all 'a' in the domain of 'f'.

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Right-Inverse of Function

A function 'g' is a right-inverse of function 'f' if applying 'f' after 'g' returns the original input. This means f(g(b)) = b for all 'b' in the domain of 'g'.

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Surjective Function

A function 'f' is surjective (onto) if every element in the codomain has at least one corresponding element in the domain. This means that for every 'b' in the codomain, there exists an 'a' in the domain such that f(a) = b.

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Injective Function

A function 'f' is injective (one-to-one) if each element in the codomain is mapped to by at most one element in the domain. This means that if f(a) = f(b), then a = b.

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Bijective Function

A function 'f' is bijective if it is both injective and surjective. This means every element in the codomain has exactly one corresponding element in the domain.

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Inverse Function

A function 'g' is the inverse of function 'f' if applying 'f' followed by 'g' (or vice-versa) returns the original input. This means that g(f(a)) = a and f(g(b)) = b for all 'a' in the domain of 'f' and 'b' in the domain of 'g'.

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How to find the inverse function?

To find the inverse of a function 'f', swap the input and output variables (x and y) and solve for the new 'y'. The resulting equation represents the inverse function, denoted as f⁻¹.

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Does every function have an inverse?

A function has an inverse (both left and right) if and only if it is bijective. This means that the function must be both injective and surjective.

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Prove Injectivity

Show that if two different inputs are given to the function, they will always result in two different outputs.

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Prove Surjectivity

Show that for any output value in the codomain, there exists at least one input value in the domain that produces it.

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How to find input for a given output in surjective function

Solve the function equation for the input variable (x) in terms of the output variable (y).

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Domain Restriction

The set of all possible input values for which the function is defined.

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Valid input for function

An input value that does not make the denominator of the function equal to zero.

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Why ad ≠ bc for injective and surjective functions

This condition ensures that the function is both injective (one-to-one) and surjective (onto), meaning it maps every input uniquely to every output without any gaps or redundancies.

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Study Notes

Algebra I - November 1, 2024

  • This document is for an Algebra I course, beginning November 1, 2024.
  • The content covers sets and functions.

Sets

  • Definition: A set is a collection of objects. Items in a set are called elements or members.
  • Notation: Capital letters (e.g., A, B, C) represent sets. Lowercase letters (e.g., a, b, c) represent elements.
  • Empty Set: The empty set (Ø) has no elements.
  • Finite Sets: A finite set has a specific, countable number of elements. The cardinality of a finite set represents the number of elements.
  • Infinite Sets: Sets with an infinite number of elements.
  • Subset: A subset (A ⊆ B) means every element in set A is also in set B. - Proper subset (A ⊂ B) is a subset where A is different from B.
  • Equal Sets: Sets are equal if they contain the exact same elements (regardless of order).
  • Notation for Describing Sets:
    • Listing elements inside braces { }.
    • Ellipses (...) to show a pattern.
    • Set builder notation, defining a set via a rule.

Set Operations

  • Union (A ∪ B): Contains all elements in either A or B or both.
  • Intersection (A ∩ B): Contains only the elements present in both sets A and B.
  • Difference (A - B): Contains elements in A but not in B. Also known as relative complement.
  • Disjoint Sets: Sets with no common elements (their intersection is empty).
  • Cartesian Product (A × B): Contains ordered pairs (a, b) where a is from set A and b is from set B.

Indexed Sets

  • Sets represented as a family of sets, labeled by an index (e.g., A₁, A₂, A₃,...). This is useful to deal with multiple sets.

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This quiz covers the fundamentals of sets and functions in Algebra I, as part of the course starting on November 1, 2024. You will explore key concepts such as definitions, notation, subsets, and types of sets, including finite and infinite sets. Test your understanding of the material presented in this segment of your algebra studies.

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