In a class of $$140$$ students numbered $$1$$ to $$140$$, all even numbered students opted mathematics course, those whose number is divisible by $$3$$ opted Physics course and those whose number is divisible by $$5$$ opted Chemistry course. Then the number of students who did not opt for any of the three courses is?
Let $$n(A)=$$ number of students opted Mathematics $$=70$$,
$$n(B)=$$ number of students opted Physics $$=46$$,
$$n(C)=$$ number of students opted Chemistry $$=28$$,
$$n(A\cap B)=23$$,
$$n(B\cap C)=9$$,
$$n(A\cap C)=14$$,
$$n(A\cap B\cap C)=4$$,
Now $$n(A\cup B\cup C)$$
$$=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B\cap C)$$
$$=70+46+28-23-9-14+4=102$$
Si number of students not opted for any course $$=$$ Total $$-n(A\cap B\cap C)$$
$$=140-102=38$$.
Of the members of three athletic teams in a school $$21$$ are in the cricket team, $$26$$ are in the hockey team and $$29$$ are in the football team. Among them, $$14$$ play hockey and cricket, $$15$$ play hockey and foot ball, and $$12$$ play foot ball and cricket. Eight play all the three games. The total number of members in the three athletic teams is
The union of the following pair of sets is: $$A = \{2, 3, 5, 6, 7\}, B = \{4, 5, 7, 8\}$$
The union of the following pair of sets is: $$C = \{a, e, i, o, u\}, D = \{a, b, c, d\}$$
Find the union of each of the following pairs of sets: (i) $$X=\left \{1,3,5\right \} Y=\left \{1,2,3\right \}$$ (ii) $$A=\{a,e,i,o,u\}, B=\{a,b,c\}$$ (iii) $$A = \{x : x \mbox{ is a natural number and multiple of } 3 \}$$. $$B = \{x : x \mbox{ is a natural number less than } 6\}$$. (iv) $$A = \{x : x \mbox { is a natural number and } 1 < x \leq 6\}$$ $$B = \{x : x \mbox{ is a natural number and} 6 < x < 10 \}$$. (v) $$A = \{1, 2, 3\}$$, $$B = \phi$$
In a committee, $$50$$ people speak French, $$20$$ speak Spanish and $$10$$ speak both Spanish and French. How many speak at least one of these two languages?
Find the smallest set $$\displaystyle Y$$ such that $$\displaystyle Y\cup \left \{ 1, 2 \right \}=\left \{ 1, 2, 3, 5, 9 \right \}$$
If A and B are two sets, then $$(A \, \cup\, B) = (A - B) \, \cup\ (B - A) \, \cup\ (A \, \cap\ B)$$