Sets, terminology and notation Flashcards
Define a set, what its contents are called and how it can be represented.
Section 1, Lesson 1
A collection of unordered, distinct, finite or infinite objects of various types.
Represented visually as a container\box, mathematically using {5, 1, 0} or symbolic, A = {5, 2, Car}
Section 1, Lesson 1
What are some of the important sets and their letter designation?
Section 1, Lesson 1
Natural numbers: Set of positive integers N = {1, 2, 3, 4, 5….}
Integers: Includes all positive and negative Z = {…. -2, -1, 0, 1, 2 …}
Rational Numbers: Any number (inc negatives) you can write as a fraction. Q = Simple {1/2, 5/8} Improper { 11/5, 15/4} Whole numbers {5, 42} Percentages {25%, 75%} Ratios {3:5, 7:10} Terminating Decimals {0.5, 0.125}
Explain Elements and Cardinality? What notation are they both expressed with?
Section 1, Lesson 1
Elements are the objects contained within the sets.
Element of: ∈ (epsilon)
Not an element of: ∉
Example: 0 ∈ A, 7 ∉ A (where A = {0, 1, 5})
Cartinality is the size of a set (number of elements)
Notation: |B| (vertical bars)
Example: If B = {square, rectangle, pentagon}, then |B| = 3
What is Set Builder Notation and how is it expressed?
Section 1, Lesson 2
A mathematical tool to describe sets concisely and flexibly, especially useful for sets following a specific pattern or rule.
{expression/rule | condition for variables}
This can be read as “the set of all elements that satisfy the expression such that the condition is true.
“Create a pile of Lego bricks. A brick goes in this pile if it’s red and it’s not damaged.”
“The set of all elements” = The pile of Lego bricks
“that satisfy the expression” = that are red
“such that” = as long as
“the condition is true” = they are not damaged
“Create a collection of items. An item is included in this collection if it matches our main description AND also meets any additional conditions we’ve set.”
Write the set of Rational Numbers (Q) using Set Builder Notation
Section 1, Lesson 2
Q = {m/n | m, n ∈ Z and n ≠ 0}
This notation can be read as: “Q is the set of all numbers of the form m/n, where m and n are integers, and n is not equal to zero.”
Q: This symbol represents the set of all rational numbers.
{}: These braces indicate that we’re defining a set.
m/n: This is the general form of a rational number, where m is the numerator and n is the denominator.
m, n ∈ Z: This means both m and n are elements of Z, where Z represents the set of all integers (…, -2, -1, 0, 1, 2, …).
and: This indicates that both conditions must be met.
n ≠ 0: This means n is not equal to zero. This is crucial because division by zero is undefined.
: This vertical bar is read as “such that” or “where”.
What is the Empty Set and how is it represented, whats its cardinality?
What is the cardinality of a set containing the Empty Set?
Section 1, Lesson 3
A special type of set with no elements inside it. Symbol: Ø or {}
Cardinality = |Ø| = 0
{Ø}| = 1
The set contains one element, which is the Empty Set itself.
What does B ⊆ A mean, what’s the difference between ⊆ and ⊂?
If|A| < |B|, can B be a subset of A?
Section 1, Lesson 5
B is a subset of A; every element in B is also in A.
⊆ means subset or equal, ⊂ means proper subset (not equal).
No, B cannot be a subset of A if it has more elements.
What is the power set of {1, 2}?
How many elements are in the power set of a set with 3 elements?
Section 1, Lesson 6
P({1, 2}) = {∅, {1}, {2}, {1, 2}}
2^3 = 8 elements
What does Aᶜ represent?
Aᶜ = U - A
{ x | x ∉ A }
The complement of A; all elements in the universal set U that are not in A.
Define A ∩ B, A ∪ B and A - B?
Section 1, Lesson 9
A ∩ B = {x | x ∈ A and x ∈ B}
The intersection of A and B; elements common to both A and B.
A ∪ B = {x | x ∈ A or x ∈ B}
The union of A and B; elements in either A or B (or both).
A - B = {x | x ∈ A and x ∉ B}
The difference of A and B; elements in A but not in B.
A ∩ B always a subset of A ∪ B?
Section 1, Lesson 9
Yes, the intersection is always a subset of the union.
What is a function?
Section 1, Lesson 11
A function is a rule that assigns each element of one set (called the domain) to exactly one element of another set (called the codomain).
