Question:
If a class of $175$ students the following data shows the number of students opting one or more subjects. Mathematics $100$, Physics $70$, Chemistry $40$, Mathematics and Physics $30$, Mathematics and Chemistry $28$, Physics and Chemistry $23$, Mathematics, Physics and Chemistry $18$. The number of students who have opted Mathematics alone is
If a class of $175$ students the following data shows the number of students opting one or more subjects. Mathematics $100$, Physics $70$, Chemistry $40$, Mathematics and Physics $30$, Mathematics and Chemistry $28$, Physics and Chemistry $23$, Mathematics, Physics and Chemistry $18$. The number of students who have opted Mathematics alone is
Updated On: May 12, 2024
- $35$
- $48$
- $60$
- $22$
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The Correct Option is C
Solution and Explanation
Let $M, P$ and $C$ be the set of students who opt Mathematics, Physics and Chemistry respectively.
Then $n(M) = 100, n(P) = 70, n(C) = 40$
$n(M \cap P) = 30, n(M \cap C) = 28, n(P \cap C) = 23,$
$n(M \cap P \cap C) = 18$
Number of students who opted Mathematics
alone = $n(M \cap P' \cap C')$
= $n(M \cap (P \cup C)')$
= $n(M) - n(M \cap (P \cup C))$
= $n(M) - n((M \cap P) \cup (M \cap C))$
= $n(M)- n(M \cap P)- n(M \cap C) + n(M \cap P \cap C) $
= $100 - 30 - 28 + 18 = 60$
Then $n(M) = 100, n(P) = 70, n(C) = 40$
$n(M \cap P) = 30, n(M \cap C) = 28, n(P \cap C) = 23,$
$n(M \cap P \cap C) = 18$
Number of students who opted Mathematics
alone = $n(M \cap P' \cap C')$
= $n(M \cap (P \cup C)')$
= $n(M) - n(M \cap (P \cup C))$
= $n(M) - n((M \cap P) \cup (M \cap C))$
= $n(M)- n(M \cap P)- n(M \cap C) + n(M \cap P \cap C) $
= $100 - 30 - 28 + 18 = 60$
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".