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?
Let F be the set of people in the committee who speak French, and S be the set of people in the committee who speak Spanish.
$$\therefore n(F)=50, n(S)=20, n(S\cap F)=10$$
We know that:
$$n(S\cup F)=n(S)+n(F)-n(S\cap F)$$
$$=20+50-10$$
$$=70-10=60$$
Thus, 60 people in the committee speak at least one of the two languages.
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: $$C = \{a, e, i, o, u\}, D = \{a, b, c, d\}$$
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?
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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 $$(A \, \cup\, B) = (A - B) \, \cup\ (B - A) \, \cup\ (A \, \cap\ B)$$