Using properties of sets prove that

For all sets $A$ and $B$, $(A\cup B)-B=A-B$

Let $T=\left\{x|\frac{x+5}{x-7}-5=\frac{4x-40}{13-x}\right\}$. Is $T$ an empty set? Justify your answer.

Let $A, B$ and $C$ be sets. Then show that $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$

Out of $100$ students; $15$ passed in English, $12$ passed in Mathematics, $8$ in Science, $6$ in English and Mathematics, $7$ in Mathematics and Science; $4$ in English and Science; $4$ in all the three. Find how many passed in English and Mathematics but not in Science

Out of $100$ students; $15$ passed in English, $12$ passed in Mathematics, $8$ in Science, $6$ in English and Mathematics, $7$ in Mathematics and Science; $4$ in English and Science; $4$ in all the three. Find how many passed in Mathematics and Science but not in English

Out of $100$ students; $15$ passed in English, $12$ passed in Mathematics, $8$ in Science, $6$ in English and Mathematics, $7$ in Mathematics and Science; $4$ in English and Science; $4$ in all the three. Find how many passed in Mathematics only

Out of $100$ students; $15$ passed in English, $12$ passed in Mathematics, $8$ in Science, $6$ in English and Mathematics, $7$ in Mathematics and Science; $4$ in English and Science; $4$ in all the three. Find how many passed in more than one subject only