An algebra problem by rounak chatterjee

Algebra Level 3

For any two real numbers $a$ and $b$ , the operation $\oplus$ defined by $a\oplus b=ab+1$ is $\text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}.$

neither commutative nor associative commutative and associative commutative but not associative associative but not commutative

1 solution

Jaydee Lucero
Nov 5, 2014

$a \oplus b=ab+1=ba+1=b \oplus a$ Therefore, the operation is commutative . However, letting $c$ be a third real number, we have $(a \oplus b) \oplus c=(ab+1) \oplus c=(ab+1)c+1=abc+c+1$ and $a \oplus (b \oplus c)=a \oplus (bc+1)=a(bc+1)+1=abc+a+1$ The equality of the two above equations only holds if $a=c$ . Thus, for all real numbers $a$ , $b$ (and $c$ ), the operation is not associative .

Did exactly the same ! Well, I think no more method to check it, so all the solvers will do the same :P

Reeshabh Kumar Ranjan - 6 years, 2 months ago

An algebra problem by rounak chatterjee

1 solution

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