Discrete Mathematics: Relations on Sets

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What property is satisfied by the relation R when demonstrated for any element a in set S?

It is reflexive. (correct)

It is asymmetric.

It is non-transitive.

It is incomplete.

Which statement correctly describes the symmetric property of relation R?

If (a, b) ∈ R, then (a, b) is not in R.

If (a, b) ∈ R, then (b, a) ∈ R. (correct)

If (b, a) ∈ R, then (b, a) does not imply (a, b).

If (b, a) ∈ R, then (a, b) is not in R.

Which of the following is true for the equivalence classes formed by the relation R?

Equivalence classes contain no elements.

There can only be one equivalence class.

Equivalence classes are disjoint sets. (correct)

Equivalence classes are overlapping.

What integers are equivalent to 0 mod 4 in the context of the relation R?

{0, 4, 8, 12, 16,...} Signup and view all the answers

Which mathematical operation defines the relation R?

Modulus. Signup and view all the answers

What conclusion can be drawn if (a, b) ∈ R and (b, c) ∈ R?

Then (a, c) ∈ R. Signup and view all the answers

Which of the following statements correctly reflects the relationship between integers 1 mod 4?

{1, 5, 9, 13, 17,...} Signup and view all the answers

What does the statement 4 | (b - a) imply about the integers a and b in relation R?

a and b are equivalent under mod 4. Signup and view all the answers

What is the set of integers equivalent to 2 mod 4?

{2, 6, 10, 14, 18,...} Signup and view all the answers

Which equivalence class contains the number 3?

P4 Signup and view all the answers

How can we represent the equivalence relation defined by (a, b) ∈ R if ab > 0?

It is both reflexive and symmetric. Signup and view all the answers

How many equivalence classes can be formed with non-negative integers under mod 4?

4 Signup and view all the answers

What is true about the set P3 = {2, 6, 10, 14, 18,...}?

It is equivalent to 2 mod 4. Signup and view all the answers

Which of the following statements about the equivalence relation (S,R) is correct?

It preserves the product of elements. Signup and view all the answers

What does the term 'equivalence class' refer to in this context?

A subset of integers where all elements have the same remainder when divided by 4. Signup and view all the answers

If you select the integer 1, which set would you be examining for integers equivalent to 1 mod 4?

{1, 5, 9, 13,...} Signup and view all the answers

What is the result of $7 + 9$ under modulo 12 arithmetic?

4 Signup and view all the answers

Which property must a relation have to be considered reflexive?

For every element $a$, $(a, a) \in R$. Signup and view all the answers

What does the notation $a \equiv b \ ( ext{mod } n)$ signify?

$a$ and $b$ give the same remainder when divided by $n$. Signup and view all the answers

Which condition is true for the symmetric property of a relation?

If $(a, b) \in R$, then $(b, a) \in R$. Signup and view all the answers

What must be true for a set of subsets to qualify as a partition of set $S$?

No two subsets can share elements. Signup and view all the answers

For which of the following values will $4 | (b - a)$ hold true when $a \equiv b \ ( ext{mod } 4)$?

$b = a + 4$ Signup and view all the answers

How does modular arithmetic differ from regular arithmetic?

It cycles back to zero after reaching a certain number. Signup and view all the answers

In a partition of set $S$, which statement is incorrect?

Two subsets can contain the same element. Signup and view all the answers

What must be demonstrated for a relation to be classified as reflexive?

Each element is related to itself. Signup and view all the answers

In the context of equivalence relations, what is the definition of symmetry?

If (a, b) relates to (c, d), then (c, d) must relate to (a, b). Signup and view all the answers

Which condition must hold for a relation to be identified as transitive?

If (a, b) is related to (c, d) and (c, d) is related to (e, f), then (a, b) must relate to (e, f). Signup and view all the answers

When determining if pairs (1, 2) and (3, 4) belong to relation R defined by a + d = b + c, which condition is satisfied?

1 + 4 = 2 + 3 Signup and view all the answers

Which statement accurately reflects the properties of equivalence relations?

An equivalence relation is always reflexive, symmetric, and transitive. Signup and view all the answers

What is the relationship between reflexivity, symmetry, and transitivity in an equivalence relation?

All three properties must exist simultaneously. Signup and view all the answers

If a relation R is defined on the integers such that (a, b),(c, d) are related if a + d = b + c, which of the following pairs are NOT in R?

(6, 2), (3, 5) Signup and view all the answers

Given the equivalence relation and the definition of R, which of the following statements is true?

Pairs (1, 2) and (2, 1) can exist in R if related. Signup and view all the answers

Which property must a relation satisfy to be classified as reflexive?

Every element is related to itself. Signup and view all the answers

In proving that a relation is symmetric, what must be shown?

If (a, b) is in R, then (b, a) is also in R. Signup and view all the answers

Which of the following relations is not symmetric based on the provided example?

R defined as (a, b) such that a ≤ b. Signup and view all the answers

To establish transitivity in a relation, which condition must hold?

(a, b) in R and (b, c) in R imply (a, c) in R. Signup and view all the answers

Which relation is proven to be an equivalence relation?

R defined as (a, b) such that a + b is even. Signup and view all the answers

What is the outcome if a relation fails any one of the properties of equivalence?

The relation cannot be classified as an equivalence relation. Signup and view all the answers

In the equivalence relation defined by R as (a, b) such that ad = bc, what is required for it to hold?

b and d must be non-zero. Signup and view all the answers

Which of the following statements is true regarding the equivalence relation?

Equivalence relations allow multiple distinct equivalence classes. Signup and view all the answers

Which property of a relation indicates that every element is related to itself?

Reflexive Signup and view all the answers

If a relation R is symmetric, which of the following statements must be true?

If (a, b) is in R, then (b, a) is also in R. Signup and view all the answers

Which of the following defines a transitive relation?

If (a, b) is in R and (b, c) is in R, then (a, c) is in R. Signup and view all the answers

What conclusion can be drawn if a relation is both antisymmetric and transitive?

The relation can only include pairs of the form (x, x). Signup and view all the answers

In the relation R defined by 'a divides b', which of the following pairs demonstrates antisymmetry?

(5, 10) and (10, 5) Signup and view all the answers

Which of the following is a characteristic of equivalence relations?

They are always reflexive and symmetric. Signup and view all the answers

Which of the following statements best describes the property of divisibility among integers?

Divisibility is always transitive. Signup and view all the answers

If the relation R defined on the set of positive integers states that (a, b) is in R if and only if a divides b, what type of relation is it?

Partially ordered relation Signup and view all the answers

Study Notes

Discrete Mathematics for Computing - Lecture Notes

Topic: Relations on Sets, Equivalence Relations, and Equivalence Classes
Date: 10/22/24

Definitions and Notations

Divisibility: A divides B (a|b) if there exists an integer s such that b = sa.
Notation: a|b
Important Note: Every integer divides 0.

Properties of Divisibility

Reflexive: a|a because a = 1 * a
Symmetric: If a|b, then b|a (Not true in general for integers.)
Transitive: If a|b and b|c, then a|c.

Worksheet Examples (Properties of Relations)

Relation: a|b (a divides b) on sets of positive integers
Reflexive: TRUE (a|a since a * 1 = a)
Symmetric: FALSE (3 | 15, but 15 does not divide 3)
Antisymmetric: If (a,b) ∈ R and (b,a) ∈ R, then a = b.
- A relation R is antisymmetric if, whenever (a,b) is in R and (b,a) is in R, then a must equal b.

Example: Proving a Relation is an Equivalence Relation Steps

Let S be the set of integers.
Let R be a relation such that (a, b) ∈ R ⇔ a+b is even.
Reflexive: (a, a) is in R since a + a = 2a is even for all a ∈ S.
Symmetric: Assume (a,b) ∈ R, then a+b is even. Then b+a is also even, so (b,a) ∈ R.
Transitive: Assume (a,b) ∈ R and (b,c) ∈ R ⇒ a+b is even and b+c is even. Then a+b+b+c = a+2b+c is even. If a+b is even and b+c is even, then a+2b+c is even, so a+c is even, which means (a,c) ∈ R.

More Example (Equivalence relation and Equivalence Classes)

Let S = Z+ and (a, b) ∈ R iff a ≤ b (a is less than or equal to b)
Reflexive: TRUE (a ≤ a)
Symmetric: FALSE (2 ≤ 3, but 3 ≤ 2 is false)
Conclusion: Not an equivalence relation

More Examples (Modular Arithmetic)

Example: 7 + 9 = 4 (mod 12). This means when you divide 7+9 by 12, the remainder is 4.
General: a = b (mod n) ⇔ n | (b - a).

Partition of a Set

A partition of a set S is a set of subsets such that (i) the union of all subsets is S. and (ii) the intersection of any two distinct subsets is empty, which means that no two overlapping pieces.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

This quiz covers key concepts related to relations on sets, specifically focusing on equivalence relations and equivalence classes. You'll explore definitions, properties of divisibility, and the conditions that determine various types of relations. Test your understanding of reflexive, symmetric, and antisymmetric properties in this engaging exercise.