Student Questions for Sets, Relations, and Functions

These questions were collected through Sli.do during the first five lectures of my Fall 2024 semester.

I have used these questions to improve my lecture notes, in terms of chains of dependency and clarity of notations, and continue to use them to refine them.

• Use of Symbols:
  • When do we use “=” versus the triple equals sign (≡) to indicate equivalence?
  • Can we use the vertical bar “|” to represent “such that” in set-builder notation?
  • I think there are two different notations for a subset. Is the subset notation underlined? It is in the lecture notes.
  • Is the subset notation with and without the line underneath really interchangeable? I thought one means the two sets can be equal.
  • Can you write the complement as ( A’ ) also?

• Subset Relationships:

  • So if ( A \subset B ) means every element of ( A ) is an element of ( B ), it does not mean that every element of ( B ) belongs to ( A )? Just a one-way relationship?
  • In this case where there is a subset, e.g., ( {7} ), can we say 7 is a subset of ( {7} )?
  • Is ( {7} ) a subset of ( X )?

• Sets Within Sets:

  • Didn’t you say an item inside a set can also be a set?
  • How many elements does ( {{}} ) have?
  • What about ( {{{}}} )?
  • What is the subset of ( {{}} )?
  • Anything not separated by commas within a set, even if it’s ( {{}} ) or ( {{{}}} ), is considered one element?
  • We can never extract 7 out of ( {{7}} ), right? ( {7} ) cannot be a subset of ( {{7}} ).
  • Once a group of elements is clumped into a subset, all of the elements are now forever bonded together? For example, in ( {1, 2, {3, 4, 5}} ), are 3, 4, 5 considered one element?

• Cardinality:

  • Does cardinality account for duplicate elements in the set? So is the cardinality of ( {1, 3, 3, 4} ) equal to 3 or 4?
  • Isn’t cardinality just the number of unique elements in a set?
  • Is the cardinality of a power set always ( 2 ) to the power of the number of elements in the previous set? Because you can be asked what is the cardinality of a power set of a power set, and so on.

• Ellipses and Patterns:
  • What if the pattern is unclear? Does the question have to state the pattern?
  • Is it possible for 4.5 to be in the set ( {1, 2, 3, \ldots, 5} )?
  • Can ellipses be used in set roster notation if we specify the conditions, especially for non-number sets like colors (e.g., colors of the rainbow)?
  • So as long as a pattern can be observed and followed, the ellipsis can be used in set roster notation?
  • When you said “between,” is it inclusive?

• Natural Numbers and Zero:

  • So 0 is part of ( \mathbb{N} ) (natural numbers), but 0 is not part of ( \mathbb{N}^* )?
  • We established that set ( \mathbb{N} ) has only numbers from 0, 1, 2, 3. For the question “Let Odd equal…,” don’t we only have two odd numbers from set ( \mathbb{N} )? That is, 1 and 3?
  • For question 5, should ( \mathbb{N} = {0, 1, 2, 3, \ldots} )?

• Rational Numbers:

  • Would 1.0 be a rational number?

• Odd Numbers and Negatives:

  • Odd numbers can be negative, right? Here, we just define them to be in the set of natural numbers.

• Venn Diagrams and Proofs:

  • In this class, can a Venn Diagram be a way of mathematical proof?

• Set Difference and Complement:

  • The lecture notes include the set difference operation; would we need to know that as well?
  • How is ( B \setminus A ) different from the complement of ( A )? Or are they the same?
  • How do we get the complement when the set ( U ) is not specified? For example, Odd’s complement: ( \mathbb{N} - \text{Odd} ), ( \mathbb{Z} - \text{Odd} ), ( \mathbb{R} - \text{Odd} ), or ( \mathbb{C} - \text{Odd} )?
  • Can you write the complement as ( A’ ) also?

• Set Builder Notation:

  • For ( A \cup B ), would " ( x ) belongs to ( A ) or ( x ) belongs to ( B )" be a way of writing it as well?
  • Isn’t ( A \cup B = {x \in U \mid x \in A \text{ or } x \in B} )?
  • For the complement, can I also write " ( x ) belongs to ( B )" too, or would that be redundant?
  • If ( U ) is not defined in a question, can we just say ( A \cup B = {x \mid x \in A \text{ or } x \in B} )?

• Associative and Commutative Laws:

  • Is ( (A \cup B) \cap C = A \cap (B \cap C) ) correctly associative?
  • Commutative property is just flipping the operands, as long as all operators are the same?
  • Commutative property versus commutative laws—are they the same?
  • Can something show both the commutative and associative laws at the same time? Because that’s the third option, right?
  • Associative means moving parentheses around each set, while commutative means flipping the sets (but both require the same operators), right?

• Non-Associative Operations:

  • Is subtraction non-associative?
  • Is division non-associative?
  • Does the absorption law apply in regular arithmetic, or is it just applied in Boolean algebra?

• Understanding Cartesian Products:
  • Shouldn’t it be ( {(1, a), (1, b), (2, a), (2, b)} )?
  • Order only matters in the parentheses then?
  • Why is ( A_1 \times A_2 \times A_3 ) not ( {((1, a), x), ((1, b), x)} )?
  • If you try to find a Cartesian product of a set produced by a Cartesian product, would you then need to nest parentheses?
  • So if you have unknown elements, does it override the rule that order doesn’t matter for sets?

• Notation and Definitions:

  • Why write ( aRb ) instead of ( (a, b) \in R )?
  • Can you have a relation over the same set?
  • Are relations always between two values?

• Properties of Relations:

  • Can a relation be both symmetric and anti-symmetric? Such as ( {(a, a), (b, b)} )?
  • Wouldn’t it not be reflexive because state A could border state B, but state C might not border state A, right?
  • You can prove symmetry with two tuples such as ( (NJ, PA) ) and ( (PA, NJ) ), right?
  • Every element is a subset of itself. So all the sets are reflexive in nature?
  • Not really. If the relation means “greater than,” thus you can never find ( x > x ). So for this condition, it is not reflexive.
  • Can you explain what symmetric and anti-symmetric are?
  • You would use three tuples to prove transitivity, like ( (a, b) ), ( (b, c) ), ( (a, c) ), right?
  • Does anti-symmetric in this case mean “if PA is next to NJ and NJ is next to PA, then it must be the case that PA = NJ” (which is not true)?
  • So it is both reflexive and symmetric.
  • Does it mean if a relation is symmetric, it must be reflexive as well?
  • Symmetry can be proven with one or two tuples, while reflexivity requires checking every element with one tuple. This makes the difference, right?
  • Would it be that if something is reflexive it must be symmetric, but not necessarily the other way around?

• Function Definitions:

  • Range is a subset of the codomain, right?
  • So we need a one-to-one matching between ( x ) and ( y ) to be a function?
  • Countries are the input and capitals are the output, as denoted by the statement “countries to capital”?

• Mapping Elements:

  • So does this mean that ( y ) will always have the same or greater number of elements than ( x )?
  • What if two different elements of ( x ) are related to the same element of ( y )?
  • So all the elements in ( X ) should be mapped to one of the elements in ( Y ), and there is no hard rule that all elements in ( Y ) should be mapped. Is that correct?
  • Can it be one-to-one when there are more elements in ( Y ) than in ( X )?

• Function Composition and Inverses:

  • Can you nest functions? In which a function ( a ) becomes an input for another function ( x ), and another function ( y ) becomes the output for that function ( x )?
  • Will we need to prove a function is bijective by first proving it is both surjective and injective in the future?
  • So if a function is injective, we say there is definitely an inverse function?

• Specific Function Examples:

  • Is the function ( f(x) = x^2 ) from the set of integers to the set of integers onto?
  • What makes it important to identify these three properties (injective, surjective, bijective)?
  • In algebra, you can have ( f(x) ) where ( x ) can equal two values. Would that be different from the discrete math context?
  • In the previous question, does it not matter that there was no arrow (no order for the pair)?

Note: Questions unrelated to the course content or administrative in nature have been omitted for clarity.