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Let S = (1, 2, 3, 4, 5) and A = S X S. Define the following relation R on A: (a, b) R (c, d) if and only if ad = bc. Show that R is an equivalence relation and compute A/R.
written 8.0 years ago by gravatar for teamques10 teamques10 ★ 68k modified 2.9 years ago by gravatar for pedsangini276 pedsangini276 • 4.8k

Mumbai University > Computer Engineering > Sem 3 > Discrete Structures

Marks: 6 Marks

Year: Dec 2015
discrete mathematics
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written 8.0 years ago by gravatar for teamques10 teamques10 ★ 68k •   modified 8.0 years ago

  1. R is an equivalence, if

    a. Reflexive:

    (a, b) R (c, d) i.e. ad=bc which is true. Hence R is reflexive.

    b. Symmetric:

    (a, b) R (c, d) $- \gt ad=bc \\ - \gt bc=ad- \gt cb=da- \gt (c, d) R (a, b)$

    Hence relation is symmetric.

    c. Transitive:

    Let (a, b) R (c, d) and (c, d) R (e, f) then ad=bc and cf=de then add ad + cf=bc + de or af=be or (a, b)R(e, f) then R is equivalent relation.

  2. For A/R

    [1]= {1, 2} = [2] [3]={3, 4}=[4].

    Hence, A/R= {[1], [3]}.
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