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Word Problems/Binary Operations

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Question
Somtimes, studying mathematics( otherwise known as scientific language) is pointless. For example, I get frustrated and exhausted when I come across a question like this: THE OPERATION * ON THE SET R OF REAL NUMBERS IS DEFINED BY A*B=(2A-1)/2 + B FOR ALL A, B BELONG TO R
1 IS THE SET R CLOSED UNDER * ?
2 IS THE OPERATION * COMMUTATIVE IN R?
3 IS THE OPERATION * ASSOCIATIVE IN R?
4 DETERMINE THE IDENTITY ELEMENT IN R
5 WHAT IS THE INVERSE OF THE ELEMENT X, IF X BELONG TO R.
Please help me solve that!

Answer
A*B = (2A-1)/2 + B ∀ A,B ∈ ℝ

(2A-1)/2 + B is real for all real values of A and B, so ℝ is closed under *.
:::::
A*B = (2A-1)/2 + B
= A - + B
= B - + A
= (2B-1)/2 + A
= B*A
* is commutative
:::::
(A*B)*C = [(2A-1)/2 + B]*C
= (2[(2A-1)/2 + B] - 1)/2 + C
= (2A-1 + 2B - 1)/2 + C
= A + B - 1 + C

A*(B*C) = (2A-1)/2 + (B*C)
= (2A-1)/2 + (2B-1)/2 + C
= (2A + 2B - 2)/2 + C
= A + B - 1 + C

(A*B)*C = A*(B*C)
* is associative
:::::
If I is the identity element, then A*I = A
(2A-1)/2 + I = A
I = A - (2A-1)/2
 = 2A/2 - (2A-1)/2
 = 1/2
:::::
I don't understand the last question.

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Janet Yang

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Word problems are my favorite type of math questions! I would not feel comfortable answering questions that require specialized knowledge (Physics, Statistics, etc.) because I have not studied these in depth.

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