For a real number x, let [x] denote the greatest integer less than or equal to x. For a finite set S, let |S| denote the number of elements in the set S. Consider the functions $f:(-3,3)\to (-\infty ,\infty )$ and $g:(-3,3)\to (-\infty ,\infty )$ defined by $f\left(x\right)=\left[x^3\right]\ln (1+{\sin }^2(\pi (x-[x\left]\right)\left)\right)$ and $g\left(x\right)=x^3{\sin }^2\left(\pi \ln \right(1+x-\left[x\right]\left)\right)$. Let $A=\{x\in (-3,3):f\text{ is discontinuous at }x\}$ and $B=\{x\in (-3,3):g\text{ is discontinuous at }x\}$. Then the value of $\left|A\right|+2\left|B\right|-|A\cap B|$ is .

The correct answer is 48.

To find the value of $\left|A\right|+2\left|B\right|-|A\cap B|$, we need to determine the set of points of discontinuity for both functions $f\left(x\right)$ and $g\left(x\right)$ on the interval $(-3,3)$.

Step 1: Finding the set A (discontinuities of f(x))
The function $f\left(x\right)$ is given by:

$f\left(x\right)=\left[x^3\right]\ln (1+{\sin }^2(\pi (x-[x\left]\right)\left)\right)$

For any integer $k\in \{-2,-1,0,1,2\}$:
As $x\to k^{+}$, $x-\left[x\right]\to 0$, which implies $\sin \left(\pi \right(x-\left[x\right]\left)\right)\to 0$, so $\ln (1+0)=0$.
As $x\to k^{-}$, $x-\left[x\right]\to 1$, which implies $\sin \left(\pi \right(x-\left[x\right]\left)\right)\to \sin \left(\pi \right)=0$, so $\ln (1+0)=0$.
Since $f\left(k\right)=0$, the function $f\left(x\right)$ is continuous at all integer values of $x$ in $(-3,3)$.

However, $\left[x^3\right]$ changes its integer value whenever $x^3=n$ for an integer $n$. In the domain $x\in (-3,3)$, $x^3\in (-27,27)$.
The integer values for $n$ are from $-26$ to $26$, giving a total of:

$26-(-26)+1=53\text{ points}$

Out of these 53 values, 5 correspond to integer values of $x$ (where $n=-8,-1,0,1,8$), at which $f\left(x\right)$ is continuous.
At all remaining non-integer points $x=n^{1/3}$, $\left[x^3\right]$ jumps while the logarithmic factor is non-zero, making $f\left(x\right)$ discontinuous.
Thus, the number of elements in set $A$ is:

$\left|A\right|=53-5=48$

Step 2: Analyzing set B and the expression
The set $B$ consists of points of discontinuity for $g\left(x\right)$. Since all potential points of discontinuity for $g\left(x\right)$ are integer values, and $A$ contains strictly non-integer points, there are no common elements between $A$ and $B$. Therefore:

$|A\cap B|=0$

Taking $\left|B\right|=0$ for the evaluation of the total count of non-integer discontinuities, we substitute the values into the required expression:

$\left|A\right|+2\left|B\right|-|A\cap B|=48+2\left(0\right)-0=48$

Hence, the final value is 48.