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Let f is a function from X to Y.  define a relation R in X given by R = {(a, b) :f(a) = f(b)}.  show tha R is an equivalence relation

    • Posted by Yatin Jindal 6 years, 10 months ago

    CBSE > Class 12 > Mathematics
    • 1 answers

    Naveen Sharma 6 years, 10 months ago

    Ans. Equivalence relation are Relations which are reflexive, transitive and symmetric.

    R = {(a, b): f(a) = f(b)}

    Check reflexive :

    Since f (a) = f (a),
    (a, a) ∈ R,
    Hence, R is reflexive.

    Check symmetric:

    If f (a) = f (b), then f (b) = f (a)
    Hence, (b, a) ∈ R.

    So, if (a, b) ∈ R
    then (b, a) ∈ R.
    R is symmetric.

    Check transitive :
    If (a, b) ∈ R
    ⇒ f(a) = f(b)    ........... (1)
     
    Also if, (b, c)∈ R
    ⇒ f(b) = f(a)  .................(2)

    From (1) & (2)
    f(a) = f(c)
    ⇒ (a, c) ∈ R,
    If (a, b) ∈ R & (b, c) ∈ R ,

    then (a, c) ∈ R

    R is transitive.

    Hence, R is an equivalence relation.

    1Thank You

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