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

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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?

The empty set is a subset of every set. Signup and view all the answers

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

A function is surjective if it has a right-inverse. Signup and view all the answers

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

The function f is bijective. Signup and view all the answers

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

If every element of A is in B and every element of B is in A. Signup and view all the answers

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

g(f(a)) = a for all a ∈ A. Signup and view all the answers

What can be inferred from the statement A ⊆ B?

All elements of A are also in B. Signup and view all the answers

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

They are identical if both exist. 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?

A is a proper subset of B. Signup and view all the answers

What is the power set of a set A?

The set of all possible subsets of A. Signup and view all the answers

Which condition guarantees that f has a right-inverse?

f is surjective. Signup and view all the answers

Which of the following statements is true regarding power sets?

If A ⊆ B, then P(A) ⊆ P(B). Signup and view all the answers

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

b is not in the image of f. Signup and view all the answers

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

Every element in A must also be an element of B. Signup and view all the answers

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

g ◦ f is surjective Signup and view all the answers

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

g ◦ f is bijective Signup and view all the answers

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

g must be surjective Signup and view all the answers

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

f must be injective 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?

a1 and a2 must be equal Signup and view all the answers

Which statement is true concerning the identity function iA?

iA(a) = a for all a ∈ A Signup and view all the answers

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

f ◦ g must equal the identity function iB 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?

a1 = a2 Signup and view all the answers

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

h(x) = h(y) implies x = y Signup and view all the answers

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

dy - b = 0 when y = ac Signup and view all the answers

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

A necessary condition for injectivity Signup and view all the answers

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

It indicates a linear relationship between x and y Signup and view all the answers

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

It keeps the denominator non-zero Signup and view all the answers

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

y cannot equal ac Signup and view all the answers

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

It implies y = x when ad ≠ bc Signup and view all the answers

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

It ensures that x and y can be uniquely determined Signup and view all the answers

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

The set of elements in the universal set U that are not in A Signup and view all the answers

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

It can produce ordered pairs by combining elements from A and B Signup and view all the answers

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

(A ∪ B)c = Ac ∩ Bc and (A ∩ B)c = Ac ∪ Bc 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|?

12 Signup and view all the answers

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

It is equal to the empty set Signup and view all the answers

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

{(1, 3), (1, 4), (2, 3), (2, 4)} Signup and view all the answers

What does the equation A ∪ Ac equal to?

U Signup and view all the answers

What property does the Cartesian product satisfy?

A × (B ∪ C) = A × B ∩ A × C Signup and view all the answers

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

$f(a)$ is in $D1 ext{ or } D2$ Signup and view all the answers

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

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

Which of the following defines the restriction of a function?

A function defined for a smaller subset of the original domain Signup and view all the answers

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

$h$ can produce outputs not in the range of $g$ Signup and view all the answers

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

Every input in $A$ maps to a unique output in $C$ Signup and view all the answers

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

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

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

$f(a)$ cannot be in either $D1$ or $D2$ Signup and view all the answers

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

It represents shared outputs from these functions Signup and view all the answers

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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Description

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.