symbols as operations

• May 29th 2012, 04:08 PM
klg
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.
• May 29th 2012, 05:13 PM
Plato
Re: symbols as operations
Quote:

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)\$
• May 29th 2012, 05:15 PM
emakarov
Re: symbols as operations
Quote:

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.
• May 29th 2012, 05:45 PM
klg
Re: symbols as operations
Thank you very much! Now this makes sense, I just hope I can apply this the next time.