GE Math 1A: Sets and Set Operations
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Questions and Answers

Which of the following mathematicians is credited with introducing the concept of sets?

  • Georg Cantor (correct)
  • Alan Turing
  • Euclid
  • Pythagoras

The set of all real numbers is denoted by the symbol ℤ.

False (B)

What is the term for a set that contains all elements of related sets under consideration?

Universal Set

A __________ is a collection of related and well-defined objects called elements.

<p>set</p> Signup and view all the answers

Match the set notation examples with their descriptions:

<p>A = {2, 4, 6, 8} = Roster Notation C = {x | 0 &lt; x &lt; 5} = Set-Builder Notation</p> Signup and view all the answers

Which of the sets below is equivalent to $D = {x | x \in ℕ, 3 < x < 7}$?

<p>{4, 5, 6} (B)</p> Signup and view all the answers

If set A is a subset of set B, then set B must be a subset of set A.

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

Express the following set in set-builder notation: The set of all even numbers greater than 0.

<p>{x | x = 2n, n ∈ ℕ}</p> Signup and view all the answers

Which of the following is NOT typically a topic covered within the course 'GE Math 1a', according to the course outline?

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

When writing a set using __________, elements are listed and separated by commas within curly braces.

<p>roster notation</p> Signup and view all the answers

Which of the following statements accurately describes an empty set?

<p>It contains no elements and is denoted by {} or ∅. (D)</p> Signup and view all the answers

If set A = {a, b, c, d}, then n(A) = 3.

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

If A is a subset of B, what must be true about the elements of A in relation to B?

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

If set A has n elements, then the number of subsets of A is given by the formula 2 to the power of ______.

<p>n</p> Signup and view all the answers

Which of the following conditions must be true for set A to be considered a proper subset of set B?

<p>Every element of A is in B, and B also contains elements not in A. (A)</p> Signup and view all the answers

The power set of a set S includes only the non-empty subsets of S.

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

Given set X = {1, 2, 3}, which of the following represents the power set P(X) correctly?

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

If set A has 3 elements, how many elements are in its power set P(A)?

<p>8</p> Signup and view all the answers

A subset that is not a proper subset must be ______ to the original set.

<p>equal</p> Signup and view all the answers

Which of the following sets is equal to $X = {x ∈ ℕ | 0 < x < 5}$?

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

Given a set $S = {a}$, the statement ${a} \subseteq P(S)$ is true.

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

For any set, the intersection of that set with its complement results in the universal set.

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

If set $A = {1, 2, 3}$ and set $B = {3, 4, 5}$, what is $A \cup B$?

<p>{1, 2, 3, 4, 5} (A)</p> Signup and view all the answers

Given the universal set $U = {1, 2, 3, 4, 5}$ and the set $A = {2, 4}$, what is $A'$?

<p>{1, 3, 5} (C)</p> Signup and view all the answers

If set $A = {a, b}$ and set $B = {1, 2}$, what is the Cartesian product $A \times B$?

<p>{(a, 1), (a, 2), (b, 1), (b, 2)} (C)</p> Signup and view all the answers

If $A={1, 2, 3, 4, 5}$ and $B={4, 5, 6, 7}$, what is $A - B$?

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

The ________ of two sets contains all the elements in both sets.

<p>union</p> Signup and view all the answers

Match the set operations with their descriptions:

<p>Union = Combines all elements from multiple sets. Intersection = Includes only the common elements between sets. Complement = Includes elements in the universal set but not in the specified set. Difference = Includes elements present in one set but not in another.</p> Signup and view all the answers

Which of the following statements about set operations is always true?

<p>$A \cup U = U$ where U is the universal set (C)</p> Signup and view all the answers

Explain the difference between the intersection and the Cartesian product of two sets.

<p>The intersection finds common elements between sets; the Cartesian product creates ordered pairs combining elements from both sets.</p> Signup and view all the answers

Flashcards

Set

A collection of related, well-defined objects called elements.

Universal Set

All elements of related sets, including its subsets.

Roster Notation

A way of listing elements separated by commas within curly braces.

Set-Builder Notation

A way of representing a set by describing the properties its elements must satisfy.

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Real numbers.

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Numbers that can be expressed as a fraction p/q.

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Whole numbers and their negatives.

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Positive whole numbers (starting from 1).

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𝕎

Non-negative integers (starting from 0).

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Element (∈)

An object that belongs to a set.

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Power Set P(S)

The set of all subsets of S.

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Union of Sets

Combines all elements from multiple sets into one set, removing duplicates.

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Intersection of Sets

Contains only the elements that are common to all sets being considered.

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Complement of a Set

Elements in the universal set U that are NOT in set A.

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Difference of Sets

Elements in A that are not in B (A - B) and vice versa (B - A).

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Cartesian Product

The set of all possible ordered pairs (a, b) where a is in A and b is in B.

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Ordered Pairs

A collection of objects where order and duplicates do matter

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Empty set

A set containing no elements

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Cardinality of a Set

The number of elements in a set. Denoted as n(A) for set A.

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Subset

A set where all its elements are also elements of another set.

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Proper Subset

A subset that is not equal to the original set.

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Power Set

The set of all possible subsets of a set, including the empty set.

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Number of Subsets

Use the formula 2^n, where n is the number of elements in the set.

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A ⊆ B

If A and B are sets, A is a subset of B (A ⊆ B) if every element in A is also in B.

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A ⊂ B

If A and B are sets, A is a proper subset of B (A ⊂ B) if every element in A is also in B, but B has at least one element not in A.

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Study Notes

  • GE Math 1a course covers topics such as the nature and language of mathematics, problem solving, biostatistics, data handling, hypothesis testing, and various statistical tests.
  • Class rules include being punctual, respectful, and attentive, as well as prohibiting the use of cellphones.
  • Mathematics uses specific symbols, syntax, and rules.
  • The learning objectives are to define sets, find subsets, perform set operations, and use Venn Diagrams.

Key Concepts of Sets

  • A set is a collection of related, well-defined objects called elements, denoted by ∈.
  • Georg Cantor introduced the word set in 1879.

Universal Set

  • The universal set contains all elements from related sets and is denoted by U.
  • R represents real numbers.
  • Q represents rational numbers.
  • Z represents integers.
  • N represents natural numbers.
  • W represents whole numbers.

Writing a Set

  • Roster notation lists elements separated by commas. Ex: A = {2, 4, 6, 8}; B = {3, 5, 7, 9}
  • Set-builder notation represents properties satisfied by the elements of a set. Ex: X = {x ∈ N|0 < x < 8}

Empty set

  • An empty set contains no elements and is denoted as {} or Ø.
  • The empty set is always a subset of any set

Cardinality of a set

  • The cardinality of set A, denoted by n(A) is the number of elements in A.
  • Ex: If A = {2,4,6,8,10}, then n(A) = 5.

Subset

  • Subsets are sets within a universal set or another set.
  • If A and B are sets, A is a subset of B (A⊆ B) if every element of A is also an element of B.
  • A⊆ B means that for all elements x ∈ A, then x ∈ B.
  • A⊆ B is read as ""A"" is a subset of ""B"".
  • A ⊈ B is read as ""A"" is not a subset of ""B"".
  • The formula 2^n determines the number of subsets in a sets with n number of elements.
    • If n(A) = 6, then 2^6 = 64 subsets.

Proper subset

  • A proper subset is any subset of a set, excluding the set itself.
  • A ⊂ B means A is a proper subset of B- every element of A is in B and at least one element of B that is not in A.

Power of a set

  • The power of a set is the set of all subsets for any given set, including the empty set.
  • For S = {a, i, r}, P(S) is the power of set S: P(S) = { ¢, {a}, {r}, {i}, {a,i}, {a, r}, {i, r}, {i,r,a}}

Operations of sets

  • The different operations of sets are:
    • Union
    • Intersection
    • Difference
    • Complement
    • Cartesian Product

Union

  • The union of two or more sets contains ALL the elements in all the sets.
  • The union of sets A and B is expressed as A U B.

Intersection

  • The intersection of two or more sets contains the common elements in all the sets.
  • The intersection of sets A and B is expressed as A ∩ B.

Complement

  • The complement of a set contains all of the elements in the universal set U that are not found in the considered set.
  • The complement of set A is denoted by A'.

Difference

  • The difference of A by B (A – B) is the set containing elements of A excluding elements found in both A and B
  • The difference of B by A (B – A) is the set containing elements of B excluding elements found in both A and B.
  • The difference of A and B vice versa.

Cartesian Products

  • Given sets A and B, the Cartesian product of A and B (A × B) is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
  • Symbolically, A×B = {(a, b) where a ∈ A, b ∈ B}.

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Description

This lesson covers the fundamentals of sets, including roster and set-builder notation. Key topics include understanding the universal set and performing set operations. Examples of sets such as Real numbers, Rational Numbers, Integers, Natural numbers, and Whole numbers are also discussed.

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