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Equivalence Relations and Partitions: Discovering Symmetry and Equivalence in Sets

Description: Equivalence Relations and Partitions: Discovering Symmetry and Equivalence in Sets
Number of Questions: 15
Created by:
Tags: set theory equivalence relations partitions symmetry
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Which of the following is an equivalence relation on the set of integers?

  1. Congruence modulo 3

  2. Less than or equal to

  3. Divisibility by 2

  4. Remainder when divided by 5

Show answer
Correct Option: A
Explanation:

Congruence modulo 3 is an equivalence relation because it satisfies the properties of reflexivity, symmetry, and transitivity.

Let R be the relation on the set of real numbers defined by xRy if and only if x - y is an integer. Is R an equivalence relation?

  1. Yes

  2. No

Show answer
Correct Option: A
Explanation:

R is an equivalence relation because it satisfies the properties of reflexivity, symmetry, and transitivity.

Let S be the relation on the set of strings defined by xSy if and only if x and y have the same length. Is S an equivalence relation?

  1. Yes

  2. No

Show answer
Correct Option: A
Explanation:

S is an equivalence relation because it satisfies the properties of reflexivity, symmetry, and transitivity.

Let T be the relation on the set of sets defined by xTy if and only if x and y have the same number of elements. Is T an equivalence relation?

  1. Yes

  2. No

Show answer
Correct Option: A
Explanation:

T is an equivalence relation because it satisfies the properties of reflexivity, symmetry, and transitivity.

What is the partition of the set {1, 2, 3, 4, 5, 6, 7, 8, 9} induced by the equivalence relation of congruence modulo 3?

  1. {{1, 4, 7}, {2, 5, 8}, {3, 6, 9}}

  2. {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}

  3. {{1, 3, 5, 7, 9}, {2, 4, 6, 8}}

  4. {{1, 2, 4, 5, 7, 8}, {3, 6, 9}}

Show answer
Correct Option: A
Explanation:

The partition of the set {1, 2, 3, 4, 5, 6, 7, 8, 9} induced by the equivalence relation of congruence modulo 3 is {{1, 4, 7}, {2, 5, 8}, {3, 6, 9}}.

What is the partition of the set {a, b, c, d, e, f} induced by the equivalence relation of having the same number of sides?

  1. {{a, b, c}, {d, e, f}}

  2. {{a, c, e}, {b, d, f}}

  3. {{a, b, d, e}, {c, f}}

  4. {{a, c, f}, {b, d, e}}

Show answer
Correct Option: A
Explanation:

The partition of the set {a, b, c, d, e, f} induced by the equivalence relation of having the same number of sides is {{a, b, c}, {d, e, f}}.

Let R be an equivalence relation on a set A. Which of the following is true?

  1. For all x in A, xRx

  2. For all x and y in A, if xRy then yRx

  3. For all x, y, and z in A, if xRy and yRz then xRz

  4. All of the above

Show answer
Correct Option: D
Explanation:

All of the above statements are true for an equivalence relation R on a set A.

Let P be a partition of a set A. Which of the following is true?

  1. The union of all the sets in P is A

  2. The intersection of any two sets in P is empty

  3. Every element of A belongs to exactly one set in P

  4. All of the above

Show answer
Correct Option: D
Explanation:

All of the above statements are true for a partition P of a set A.

Which of the following is an example of a partition of the set {1, 2, 3, 4, 5, 6}?

  1. {{1, 2, 3}, {4, 5, 6}}

  2. {{1, 3, 5}, {2, 4, 6}}

  3. {{1, 2}, {3, 4}, {5, 6}}

  4. {{1, 2, 3, 4}, {5, 6}}

Show answer
Correct Option: A
Explanation:

{{1, 2, 3}, {4, 5, 6}} is an example of a partition of the set {1, 2, 3, 4, 5, 6}.

Which of the following is an example of an equivalence relation on the set {1, 2, 3, 4, 5, 6}?

  1. Congruence modulo 2

  2. Less than or equal to

  3. Divisibility by 3

  4. Remainder when divided by 4

Show answer
Correct Option: A
Explanation:

Congruence modulo 2 is an example of an equivalence relation on the set {1, 2, 3, 4, 5, 6}.

Let R be an equivalence relation on a set A. Which of the following is true?

  1. The equivalence class of an element x in A is the set of all elements in A that are related to x by R

  2. The equivalence class of an element x in A is the set of all elements in A that are not related to x by R

  3. The equivalence class of an element x in A is the set of all elements in A that are equal to x

  4. None of the above

Show answer
Correct Option: A
Explanation:

The equivalence class of an element x in A is the set of all elements in A that are related to x by R.

Let P be a partition of a set A. Which of the following is true?

  1. Each set in P is an equivalence class of some equivalence relation on A

  2. The union of all the sets in P is A

  3. The intersection of any two sets in P is empty

  4. All of the above

Show answer
Correct Option: D
Explanation:

All of the above statements are true for a partition P of a set A.

Which of the following is an example of a partition of the set {a, b, c, d, e, f}?

  1. {{a, b, c}, {d, e, f}}

  2. {{a, c, e}, {b, d, f}}

  3. {{a, b}, {c, d}, {e, f}}

  4. {{a, c, f}, {b, d, e}}

Show answer
Correct Option: A
Explanation:

{{a, b, c}, {d, e, f}} is an example of a partition of the set {a, b, c, d, e, f}.

Which of the following is an example of an equivalence relation on the set {a, b, c, d, e, f}?

  1. Congruence modulo 2

  2. Less than or equal to

  3. Divisibility by 3

  4. Remainder when divided by 4

Show answer
Correct Option: A
Explanation:

Congruence modulo 2 is an example of an equivalence relation on the set {a, b, c, d, e, f}.

Let R be an equivalence relation on a set A. Which of the following is true?

  1. The equivalence class of an element x in A is the set of all elements in A that are related to x by R

  2. The equivalence class of an element x in A is the set of all elements in A that are not related to x by R

  3. The equivalence class of an element x in A is the set of all elements in A that are equal to x

  4. None of the above

Show answer
Correct Option: A
Explanation:

The equivalence class of an element x in A is the set of all elements in A that are related to x by R.

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