Question

Which of the following are binary operations on S = {x ∈ R | x > 0}? Justify your answer. i) ii) The operation dened by xy=|ln(xy)| where lnx is the natural logarithm. The operation dened by xy=x2+y3

Answer 1

Answer on Question #80842 – Math – Linear Algebra

Question

Which of the following are binary operations on $S = \{x \in \mathbb{R} \mid x > 0\}$? Justify your answer.

i) The operation defined by $x * y = |\ln(xy)|$ where $\ln x$ is the natural logarithm.

ii) The operation defined by $x * y = x^2 + y^3$.

Solution

An operation is binary if the result of performing the operation on a pair of elements of $S$ is again an element of $S$.

Here both operations i) and ii) require 2 arguments (marked as $x$ and $y$).

i) If we take $x = y = 1$, then

$$x * y = | \ln (1 - 1) | = 0.$$

In other words, $x * y \notin S$ if $x \in S, y \in S$.

Therefore, the operation defined by $x * y = |\ln(xy)|$ is not a binary operation.

ii) The operation defined by $x * y = x^2 + y^3$ is binary operation because $x^2 + y^3 > 0$ if $x > 0, y > 0$. In other words, $x * y \in S$ if $x \in S, y \in S$.

Answer: only ii) is a binary operation.

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