practice test prob: Which is the identity element?

lyzhou1990

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I got this problem from a practice test and had no idea how to solve it. (In fact, I'm not even sure if I'm putting it in the right place...) Anyway, the problem is below. I'd appreciate it if someone could answer this problem and give an explanation, too.

36. The operation given by (a, b)[diamond](c, d) = (a + bc, bd) is defined on the set of ordered pairs with nonzero second elements. Which of the following is the identity element for this operation?

A) (0, 0)
B) (1, 1)
C) (1, 0)
D) (0, 1)
E) (0, -1)

Thanks again.
 
One method of solution would be to plug the various answers (and a generic ordered pair, such as "(m, n)") into the formula for the "diamond" operator, and see which one acts as the identity.

Another method would be to take the formula, as provided, and set it equal to the original ordered pair (a, b), solving for the values of c and d. Then solve the other way (with (c, d) being the non-identity in the formula), and solving for a and b. Verify that you got the same thing.

What have you tried? Where are you stuck?

Thank you.

Eliz.
 
Thanks for the help.

My major problem is that I don't understand what it means by "nonzero second element". Does it mean that in (a,b) and (c,d) that b and d are nonzero, or does it mean that in (a,b)[diamond](c,d) that (c,d) is somehow nonzero?

And also, is there a way of figuring out the answer without plugging in numbers?

Thanks.
 
lyzhou1990 said:
My major problem is that I don't understand what it means by "nonzero second element". Does it mean that in (a,b) and (c,d) that b and d are nonzero, or does it mean that in (a,b)[diamond](c,d) that (c,d) is somehow nonzero?.
It means that in (a,b) and (c,d) that b and d are nonzero.
So that rules options A) & C).
Here is an example: (x,y)\(\displaystyle \diamond\)(0,-1)=(x+0(y),y(-1))=(x,-y).
 
Hello, lyzhou1990!

36. The operation given by \(\displaystyle (a,\,b)\,\diamond\,(c,\,d)\;=\;(a + bc,\,bd)\)
is defined on the set of ordered pairs with nonzero second elements.
Which of the following is the identity element for this operation?

\(\displaystyle A)\;(0,\,0)\;\;\;\;B)\;(1\, 1)\;\;\;\;C)\;(1,\,0)\;\;\;\;D)\;(0,\,1)\;\;\;\;E) (0,\,-1)\)

The identity element is \(\displaystyle (x,\,y)\) where: \(\displaystyle \,(a,\,b)\,\diamond\,(x,\,y)\;=\;(a,\,b)\)


By definition: \(\displaystyle (a,\,b)\,\diamond\,(x,\,y)\:=\:(a+bx,\,by)\)

Hence, we want: \(\displaystyle \,(a+bx,\,by)\:=\:(a,\,b)\)

Two ordered pairs are equal if their corresponding components are equal.

So we have: \(\displaystyle \:\begin{array}{ccc}a \,+\,bx\:=\:a \\ . \\ by \:=\:b\end{array}\;\;\Rightarrow\;\;\begin{array}{ccc}x\,=\,0\\ . \\ y\,=\,1\end{array}\)


Therefore, the identity element is: \(\displaystyle \,(0,\,1)\;\) . . . answer (D)

 
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