Question

For all numbers a and b, let a(.)b be defined by a (.) b = ab + a + b. For all numbers x, y, and z, which of the following must be true?
#1. x(.)y = y (.) x
#2. (x-1) (.) (x+1) = (x(.)x)-1
#3. x(.) (y+z) = (x (.) y ) + (x (.) z)

***the places that i have the "(.)" symbol, its because there is a dot with a circle around it in my book, ive never seen it before!

Answer 1

ah you are checking to see if it is commutative. so lets do it

Answer 2

I've never studies that, that i recall, but its gonna be on my SAT

Answer 3

mridrik: Will someone please look at my last question, it has to deal with cubes and I need help from someone very smart.

Answer 4

we want to see if
\[x\oplus y=y\oplus x\]so don't be confused by this, just go right to the definition and see if it is true or not. i.e. work from the definition of the operation

Answer 5

the left hand side of
\[x\oplus y \] is
\[x\oplus y = xy + x + y\] and the right hand side is
\[y\oplus x = yx +y+x\] and now the question is are they equal? so we see that
\[xy+x+y=yx+y+x\] because both multiplication and addition are commutative, and therfore
\[x\oplus y = y\oplus x\]

Answer 6

by the way it is clear ( i hope) that the symbol used is not important. it is just a definition for the operation. they could have used any symbol. i use
\[\oplus\] but i could have use
\[*\] or
\[\heartsuit \] or anything

Answer 7

ready for the next one?