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rishabh.mission

  • 2 years ago

Let A = Q x Q , Q being the set of rationals . Let ‘*’ be a binary operation on A , defined by (a, b) * (c , d) = ( ac , ad + b) . Show that (i) ‘*’ is not commutative (ii) ‘*’ is associative (iii The Identity element w.r.t ‘*’ is ( 1 , 0)

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  1. rishabh.mission
    • 2 years ago
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    help me plzz

  2. sirm3d
    • 2 years ago
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    (i) (a,b) * (c,d) = (ac, ad + b) (c,d) * (a,b) = (ca,cb + d) (ac, ad + b) \(\neq\) (ca, cb + d) therefore it is not commutative

  3. rishabh.mission
    • 2 years ago
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    thanks bro and how i solve (ii) part?

  4. sirm3d
    • 2 years ago
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    compute (a,b) * ((c,d) * (e,f)) compute ((a,b) * (c,d)) * (e,f) compare the two ordered pairs if they are equal

  5. sirm3d
    • 2 years ago
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    (a,b) * ((c,d) * (e,f)) = (a,b) * (ce, cf + d) = (ace, a(cf + d) + b) ((a,b) * (c,d)) * (e,f) = (ac, ad + b) * (e,f) = (ace, acf + (ad + b)) the two ordered pairs are the same, therefore * is associative

  6. rishabh.mission
    • 2 years ago
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    thnanks again @sirm

  7. sirm3d
    • 2 years ago
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    yw

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