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Math Help - Relations

  1. #1
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    Relations

    question 1
    Let A = (1, 2, 3, 4) Let R be a relation on A^2 defined as
    R = ((1; 1); (2; 2); (3; 3); (4; 4); (2; 3); (4; 3)). Determine if R is an equivalence relation.


    question 2
    Prove or disprove that ~ is an equivalence relation:
    (a) Let N be the set of non-negative integers. A relation  ~ on the set N as follows:
    a ~ b if and only if a + b is an even integer.
    (b) A relation ~ is defined on the set of integers as x ~ y <---> x = y:
    (c) A relation ~ is defined on the set Z by x ~ y <---> x = ky ; k ∈ R.
    (d) A relation ~ is defind on Z as x ~ y if and only if x <= y
    (e) Let R+ be the set of positive real numbers. Let ~ be a relation on R+, dened as
    a ~ b <---> a/b = 2^k, k ∈ Z
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  2. #2
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    Re: Relations

    Hey beast.

    Hint: Equivalence relations have the property [a] = {b in X such that a ~ b}. Take a look at the wiki entry (it has a similar example to yours):

    Equivalence relation - Wikipedia, the free encyclopedia
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  3. #3
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    Re: Relations

    Hey, chiro,

    The equivalence class [a] is usually not even defined for non-equivalence relations.
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  4. #4
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    Re: Relations

    An equivalence relation must satisfy
    1) reflexive: if a is in the set then (a, a) is in the relation.
    2) symmetric: if (a, b) is in the relation, then (b, a) is also in the relation.
    3) transitive: if (a, b) is in the relation and (b, c) is in the relation, then (a, c) is in the relation.

    Are those true for these relations?
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