Question

Which of the following binary operations defined on the set of real numbers is not associative?

• A. Addition
• B. Subtraction
• C. Multiplication
• D. Division

Answer 1

Answer: (B), (D)

Division

A binary operation ${}^{\ast }$ is associative if a ${}^{\ast }$ (b ${}^{\ast }$ c) = (a ${}^{\ast }$ b) ${}^{\ast }$ c. Addition and multiplication are associative because a(bc) = (ab)c = abc and a+(b+c) = (a+b)+c. But we can see that subtraction and division are not associative because a-(b-c) is not equal to (a-b)-c and $\frac{a}{\left(\frac{b}{c}\right)}$ is not equal to $\frac{\left(\frac{a}{b}\right)}{c}$. So two of the given functions are not associative.