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Questions and Answers
If the relation R on the set of all integers is defined as x ≠y, which of the following properties does R satisfy?
If the relation R on the set of all integers is defined as x ≠y, which of the following properties does R satisfy?
- It is transitive.
- It is reflexive.
- It is symmetric. (correct)
- It is antisymmetric.
If the relation R is defined as x ≡ y (mod 7) on the set of all integers, which of the following properties does R satisfy?
If the relation R is defined as x ≡ y (mod 7) on the set of all integers, which of the following properties does R satisfy?
- It is symmetric and transitive. (correct)
- It is asymmetric.
- It is reflexive.
- It is not reflexive.
How many different relations are there from a set with m elements to a set with n elements?
How many different relations are there from a set with m elements to a set with n elements?
- m^n
- m × n
- 2^(m × n) (correct)
- m + n
Let R be the relation R = {(a, b) | a divides b} on the set of positive integers. What is R^(-1)?
Let R be the relation R = {(a, b) | a divides b} on the set of positive integers. What is R^(-1)?
Let f be a one-to-one correspondence from A to B. Let R be the relation that equals the graph of f. Which of the following is true about R?
Let f be a one-to-one correspondence from A to B. Let R be the relation that equals the graph of f. Which of the following is true about R?
Give an example of a relation on a set that is both symmetric and antisymmetric.
Give an example of a relation on a set that is both symmetric and antisymmetric.
What do you obtain when you apply the selection operator sC, where C is the condition (Project=2)∧(Quantity≥ 50), to the database in Table 10?
What do you obtain when you apply the selection operator sC, where C is the condition (Project=2)∧(Quantity≥ 50), to the database in Table 10?
What is the result of applying the projection P2,3,5 to the 5-tuple (a, b, c, d, e)?
What is the result of applying the projection P2,3,5 to the 5-tuple (a, b, c, d, e)?
Which projection mapping is used to delete the first, second, and fourth components of a 6-tuple?
Which projection mapping is used to delete the first, second, and fourth components of a 6-tuple?
What are the operations that correspond to the query expressed using the SQL statement SELECT Supplier FROM Part_needs WHERE 1000 ≤ Part_number ≤ 5000?
What are the operations that correspond to the query expressed using the SQL statement SELECT Supplier FROM Part_needs WHERE 1000 ≤ Part_number ≤ 5000?
Which of the following relations on {0, 1, 2, 3} are equivalence relations?
Which of the following relations on {0, 1, 2, 3} are equivalence relations?
What is the output of the query given the database in Table 9 as input?
What is the output of the query given the database in Table 9 as input?
What is the inverse relation R−1 of R = {(a, f(a)) | a ∈ A}?
What is the inverse relation R−1 of R = {(a, f(a)) | a ∈ A}?
What is the composition of R2 and R6, denoted by R2 ⊕ R6?
What is the composition of R2 and R6, denoted by R2 ⊕ R6?
How many relations are there on the set {a, b, c, d} that contain the pair (a, a)?
How many relations are there on the set {a, b, c, d} that contain the pair (a, a)?
What is the error in the 'proof' of the 'theorem' that a symmetric and transitive relation R is reflexive?
What is the error in the 'proof' of the 'theorem' that a symmetric and transitive relation R is reflexive?
What is the relation R3 ∪ R6 on the set {a, b, c, d}?
What is the relation R3 ∪ R6 on the set {a, b, c, d}?
What is the representation of the relation R5 = {(a, b) ∈ R2 | a = b} as a matrix on {1, 2, 3}?
What is the representation of the relation R5 = {(a, b) ∈ R2 | a = b} as a matrix on {1, 2, 3}?
What is the characteristic of a relation R on a set A if the matrix representing R has no nonzero entries on the main diagonal?
What is the characteristic of a relation R on a set A if the matrix representing R has no nonzero entries on the main diagonal?
What is the matrix representing the relation $R^{-1}$, where R is a relation on a finite set A?
What is the matrix representing the relation $R^{-1}$, where R is a relation on a finite set A?
If a relation R on a set A is represented by a matrix with only one nonzero entry, what can be said about the relation?
If a relation R on a set A is represented by a matrix with only one nonzero entry, what can be said about the relation?
What is the complement of a relation R on a set A, represented as a matrix?
What is the complement of a relation R on a set A, represented as a matrix?
If a relation R on a set A is represented by a matrix with all nonzero entries on the main diagonal, what can be said about the relation?
If a relation R on a set A is represented by a matrix with all nonzero entries on the main diagonal, what can be said about the relation?
What is the number of nonzero entries in the matrix representing the relation R on A = {1, 2, 3,..., 100} consisting of the first 100 positive integers, if R is {(a, b) | a > b}?
What is the number of nonzero entries in the matrix representing the relation R on A = {1, 2, 3,..., 100} consisting of the first 100 positive integers, if R is {(a, b) | a > b}?
Let R be the relation {(a, b) | a and b are the same age} on the set of humans. What is the symmetric closure of R?
Let R be the relation {(a, b) | a and b are the same age} on the set of humans. What is the symmetric closure of R?
Let R be the relation {(a, b) | a divides b} on the set of integers. What is the reflexive closure of R?
Let R be the relation {(a, b) | a divides b} on the set of integers. What is the reflexive closure of R?
Let R be the relation {(a, b) | a and b have met} on the set of humans. Is R an equivalence relation?
Let R be the relation {(a, b) | a and b have met} on the set of humans. Is R an equivalence relation?
What is the congruence class [2]5?
What is the congruence class [2]5?
Which of the following collections of subsets are partitions of {1, 2, 3, 4, 5, 6}?
Which of the following collections of subsets are partitions of {1, 2, 3, 4, 5, 6}?
Let R be a relation on a finite set. How can the directed graph representing the reflexive closure of R be constructed from the directed graph of R?
Let R be a relation on a finite set. How can the directed graph representing the reflexive closure of R be constructed from the directed graph of R?
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Study Notes
Relations in Discrete Structures
- A relation R from a set A to a set B is a subset of the Cartesian product A × B.
Properties of Relations
- Reflexive: a relation R is reflexive if for every a in A, (a, a) ∈ R.
- Irreflexive: a relation R is irreflexive if for no a in A, (a, a) ∈ R.
- Symmetric: a relation R is symmetric if for every a, b in A, (a, b) ∈ R implies (b, a) ∈ R.
- Antisymmetric: a relation R is antisymmetric if for every a, b in A, (a, b) ∈ R and (b, a) ∈ R implies a = b.
- Asymmetric: a relation R is asymmetric if it is not symmetric.
- Transitive: a relation R is transitive if for every a, b, c in A, (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R.
Examples of Relations
- a) {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)}: not reflexive, not symmetric, transitive.
- b) {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}: reflexive, symmetric, transitive.
- c) {(2, 4), (4, 2)}: not reflexive, symmetric, not transitive.
- d) {(1, 2), (2, 3), (3, 4)}: not reflexive, not symmetric, transitive.
- e) {(1, 1), (2, 2), (3, 3), (4, 4)}: reflexive, not symmetric, transitive.
- f) {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}: not reflexive, symmetric, not transitive.
Equivalence Relations
- An equivalence relation is a relation that is reflexive, symmetric, and transitive.
- Examples of equivalence relations:
- {(0, 0), (1, 1), (2, 2), (3, 3)}.
- {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)}.
- {(0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3)}.
Operations on Relations
- Union: R ∪ S = {(a, b) | (a, b) ∈ R or (a, b) ∈ S}.
- Intersection: R ∩ S = {(a, b) | (a, b) ∈ R and (a, b) ∈ S}.
- Difference: R - S = {(a, b) | (a, b) ∈ R and (a, b) ∉ S}.
- Complement: R' = {(a, b) | (a, b) ∉ R}.
- Symmetric difference: R ⊕ S = (R ∪ S) - (R ∩ S).
Matrix Representation of Relations
- A relation R on a set A can be represented as a matrix, where the entry at row i and column j is 1 if (i, j) ∈ R, and 0 otherwise.
- The matrix representing the relation R can be used to determine whether R is reflexive, irreflexive, symmetric, antisymmetric, or transitive.
Inverse Relations
- The inverse relation R^(-1) of a relation R is defined as R^(-1) = {(b, a) | (a, b) ∈ R}.
- Example: R = {(a, b) | a > b}, then R^(-1) = {(a, b) | a < b}.
Equivalence Classes
- The equivalence class [a] of an element a with respect to an equivalence relation R is the set of all elements b such that (a, b) ∈ R.
- Example: R = {(a, b) | a ≡ b (mod 5)}, then [2] = {2, 7, 12, ...}.
Partitions
- A partition of a set A is a collection of non-empty subsets of A such that every element of A is in exactly one subset.
- Example: {{1, 2}, {3, 4}, {5, 6}} is a partition of {1, 2, 3, 4, 5, 6}.
Reflexive and Symmetric Closures
- The reflexive closure of a relation R is the smallest reflexive relation that contains R.
- The symmetric closure of a relation R is the smallest symmetric relation that contains R.
- Example: R = {(a, b) | a ≠b}, then the reflexive closure of R is R ∪ {(a, a) | a ∈ A}.
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