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Math Help - symbols as operations

  1. #1
    klg
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    symbols as operations

    the question is as follows:
    For all numbers x and y, let the operation ■ be defined by x ■ y = xy-y. If a and b are positive integers, which of the following can be equal to zero?

    I. a■b
    II. (a+b)■b
    III. a■(a+b)

    does one distribute choices II and III to a■b + b■b and a■a + a■b respectively? How does that help?
    There is just something about the use of symbols that has not registered yet. Thanks.
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  2. #2
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    Re: symbols as operations

    Quote Originally Posted by klg View Post
    the question is as follows:
    For all numbers x and y, let the operation ■ be defined by x ■ y = xy-y. If a and b are positive integers, which of the following can be equal to zero?
    I. a■b
    II. (a+b)■b
    III. a■(a+b)
    I~a \bullet b = ab - b

    II~(a + b) \bullet b = (a + b)b - b

    III~a \bullet (a + b) = a(a + b) - (a + b)
    Thanks from klg
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  3. #3
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    Re: symbols as operations

    Quote Originally Posted by klg View Post
    does one distribute choices II and III to a■b + b■b and a■a + a■b respectively?
    Let's see. By definition, (x + y) ■ z = (x + y)z - z = xz + yz - z, while x ■ z + y ■ z = xz - z + yz - z = xz + yz - 2z. So, ■ does not distribute over + on positive integers.

    A binary operator is just a function of two arguments. Expand ■ according to its definition in each expression so that you only have plus and times left.
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  4. #4
    klg
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    Re: symbols as operations

    Thank you very much! Now this makes sense, I just hope I can apply this the next time.
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