Binary Operations

[jessica098]

I'm having trouble proving the following:

For all x,y ? R, define x*y to be x*y = x+y-xy.

Proposition: The binary operation * as defined above is an associative operation on R.

So far this is what I have.

Proof: Let x,y,z?R. We will prove that (x *y)*z= x *(y *z). We know that addition and multiplication of real numbers is associative, so

(x *y)*z=(x+y-xy)*z

Can anyone help me finish this proof to show that (x *y)*z= x *(y *z)?

Thanks!

 

[Unco]

I'm having trouble proving the following:

For all x,y ? R, define x*y to be x*y = x+y-xy.
...
(x *y)*z=(x+y-xy)*z

Can anyone help me finish this proof to show that (x *y)*z= x *(y *z)?

Now apply the definition of * to (x+y-xy)*z, Jessica.

Remember that you can also expand the right-hand side of (x *y)*z= x *(y *z) so you know where you're going.

 

[soroban]

Hello, jessica098!

\(\displaystyle \text{For all }x,y \in R}\text{, define: }\:x*y \:=\:x+y-xy\)

\(\displaystyle \text{Prove: the binary operation }*\text{ is associative.}\)

\(\displaystyle \text{The definition of }*\text{ is: } \:x*y \:=\:x+y-xy\)
. . \(\displaystyle \text{(Add }x\text{ and }y\text{ and subtract their product.)}\)


\(\displaystyle \text{We want to show that: }\;a*(b*c) \:=\a*b)*c\)


\(\displaystyle a*(b*c) \;=\;a*(b+c - bc)\)

. . . . . . .\(\displaystyle = \;a + (b+c-bc) - a(b+c-bc)\)

. . . . . . .\(\displaystyle = \;a + b + c - ab - bc - ac + abc\)


\(\displaystyle (a*b)*c \;=\;(a + b - ab)*c\)

. . . . . . .\(\displaystyle = \;(a + b - ab) + c - (a+b-ab)c\)

. . . . . . .\(\displaystyle = \;a + b + c - ab - bc - ac + abc\)


\(\displaystyle \text{Therefore: }\;a*(b*c) \;=\;(a*b)*c\) . . . The operation is associative.