Question

If $*$ is an operation such that for any two integers $a$ and $b$

$a*b=a-(-b)+(-3)$, find the value of $(-4)*(-5)\mathrm{and}(-5)*(-4)$

Is $a*b=b*a?$

Answer 1

Step 1: Finding the value of $(-4)*(-5)$

We have $a*b=a-(-b)+(-3)$

i.e.,

$(-4)*(-5)=(-4)-\left[-(-5)\right]+(-3)$

If the given numbers are of like signs, let's add the absolute values directly and give the common sign for the result.

$(-4)-\left[5\right]+(-3)=-12$ $\left\{Since-(-5)=+5\right\}$

Step 2: Finding the value of $(-5)*(-4)$

$\begin{array}{rcl}(-5)*(-4)& =& (-5)-\left[-(-4)\right]+(-3)\\ & =& (-5)-\left[4\right]+(-3)=-12\end{array}$ $\left\{Since-(-4)=+4\right\}$

From both the calculations, we get the value of $-12$.

Hence, we can conclude that $a*b=b*a$