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Equivalence Relations and Partitions: Discovering Symmetry and Equivalence in Sets

Equivalence Relations and Partitions: Discovering Symmetry and Equivalence in Sets

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

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
Question 2 Multiple Choice (Single Answer)

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
Question 3 Multiple Choice (Single Answer)

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
Question 4 Multiple Choice (Single Answer)

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
Question 5 Multiple Choice (Single Answer)

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}}
Question 6 Multiple Choice (Single Answer)

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}}
Question 7 Multiple Choice (Single Answer)

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
Question 8 Multiple Choice (Single Answer)

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
Question 9 Multiple Choice (Single Answer)

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}}
Question 10 Multiple Choice (Single Answer)

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
Question 11 Multiple Choice (Single Answer)

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
Question 12 Multiple Choice (Single Answer)

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
Question 13 Multiple Choice (Single Answer)

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}}
Question 14 Multiple Choice (Single Answer)

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
Question 15 Multiple Choice (Single Answer)

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
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