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Thread: equivalence class help :(

  1. #1
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    equivalence class help :(

    i just can't comprehend what an equivalence class is. i can't even get started on this problem. maybe someone can help me atleast get started?

    here's the problem

    let A = {1,2,3,4,5,6} and S = power set of A

    a) for a,b is an element of S, define a~b if a and b have the same number of elements. prove that ~ defines an equivalence relation on S.

    b)how many equivalence class are there? list one element from each equivalence class.

    i understand what an equivalence relation is. my professor gave me some hints but i can't draw anything from them.

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

    i understand that part, but then how would i do part b?
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  2. #2
    MHF Contributor Matt Westwood's Avatar
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    First you need to establish what a relation on a set $\displaystyle S$ is.

    Given the set of all ordered pairs taken from $\displaystyle S \times S$, a relation is any subset $\displaystyle R$ of this set.

    If $\displaystyle (x, y) \in R$ we can write $\displaystyle x R y$ and say "x is related to y by R".

    An equivalence relation is a relation which is:

    a) Reflexive: $\displaystyle \forall a \in S: (a, a) \in R$. All elements are related to themselves.

    b) Symmetric: $\displaystyle x R y \implies y R x$. If one element is related by R to another one, then the other one is likewise related to the first.

    c) Transitive: $\displaystyle x R y, y R z \implies x R z$.

    Examples:

    $\displaystyle =$ is an equivalence relation trivially.

    $\displaystyle <$ is not an equivalence because $\displaystyle a < a$ is always false and $\displaystyle a < b$ means it is not the case that $\displaystyle b < a$.

    Recommend you revise your work on relations, then you should be able to get a better handle on how an equivalence relation works.
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  3. #3
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    Quote Originally Posted by glopez09 View Post
    let a = {1,2,3,4,5,6} and s = power set of a

    a) for a,b is an element of s, define a~b if a and b have the same number of elements. Prove that ~ defines an equivalence relation on s.

    B)how many equivalence class are there? List one element from each equivalence class.
    But then how would i do part b?
    $\displaystyle [\emptyset] ,[\{1\}],[\{1,2\}],[\{1,2,3\}],[\{1,2,3,4\}],[\{1,2,3,4,5\}],[\{1,2,3,4,5,6\}]$
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