1. ## 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.

2. ## Re: symbols as operations

Originally Posted by klg
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)
$\displaystyle I~a \bullet b = ab - b$

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

$\displaystyle III~a \bullet (a + b) = a(a + b) - (a + b)$

3. ## Re: symbols as operations

Originally Posted by klg
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.

4. ## Re: symbols as operations

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