Question:
Let A, B, C be finite sets. Suppose that $n (A) = 10, n (B) = 15, n (C) = 20, n (A\cap B) = 8$ and $n (B\cap C) = 9$. Then the possible value of $n (A\cup B \cup C)$ is
Let A, B, C be finite sets. Suppose that $n (A) = 10, n (B) = 15, n (C) = 20, n (A\cap B) = 8$ and $n (B\cap C) = 9$. Then the possible value of $n (A\cup B \cup C)$ is
Updated On: Jul 5, 2022
- 26
- 27
- 28
- Any of the three values 26, 27, 28 is possible
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
We have
$n \left(A \cup B \cup C\right) = n \left(A\right) + n \left(B\right) + n \left(C\right) -$
$n \left(A \cap B\right) - n\left(B\cap C\right) - n \left(C \cap A\right) + n \left(A\cap B \cap C\right)$
$= 10 +15 + 20 - 8 - 9 - n \left(C \cap A\right) + n \left(A \cap B \cap C\right)$
$= 28 - n\left(C \cap A\right) - n \left(A \cap B \cap C\right)\quad ...\left(i\right)$
Since $n \left(C \cap A\right) \ge n \left(A \cap B \cap C\right)$
We have $n \left(C \cap A\right) - n \left(A \cap B \cap C\right) \ge 0\quad...\left(ii\right)$
From $\left(i\right)$ and $\left(ii\right)$
$n \left(A \cup B \cup C\right) \le 28\quad ...\left(iii\right)$
Now, $n\left(A \cup B\right) = n \left(A\right) +n \left(B\right) - n \left(A \cap B\right)$
$= 10 + 15 - 8 = 17$
and $n \left(B \cup C\right) = n \left(B\right) + n \left(C\right) - n \left(B \cap C\right)$
$= 15 + 20 - 9 = 26$
Since, $n \left(A \cup B \cup C\right) \ge n \left(A\cup C\right)$ and
$n \left(A\cup B\cup C\right) \ge n \left(B\cup C\right)$, we have
$n \left(A\cup B\cup C\right) \ge 17$ and $n \left(A\cup B\cup C\right) \ge 26$
Hence $n \left(A\cup B\cup C\right) \ge 26 \quad...\left(iv\right)$
From $\left(iii\right)$ and $\left(iv\right)$ we obtain
$26 \le n \left(A\cup B\cup C\right) \le 28$
Also $n \left(A\cup B\cup C\right)$ is a positive integer
$\therefore n\left(A\cup B\cup C\right) = 26$ or $27$ or $28$
Was this answer helpful?
0
0
Top Questions on Sets
- The equation of a common tangent to the parabolas \( y = x^2 \) and \( y = -(x - 2)^2 \) is:
- Let $A = \{2, 3, 4, 5, \ldots, 16, 17, 18\}$. Let $R$ be the relation on the set $A$ of ordered pairs of positive integers defined by $(a, b) R (c, d)$ if and only if $ad = bc$ for all $(a, b), (c, d) \in A \times A$. Then the number of ordered pairs of the equivalence class of $(3, 2)$ is:
- If \( f(x) = \frac{4x + 5}{6x - 4}, \, x \neq \frac{2}{3} \) and \( (fof)(x) = g(x) \), where \( g : \mathbb{R} - \left\{ \frac{2}{3} \right\} \rightarrow \mathbb{R} - \left\{ \frac{2}{3} \right\} \), then \( (gogogog)(4) \) is equal to
- If \[ S(x) = (1 + x) + 2(1 + x)^2 + 3(1 + x)^3 + \ldots + 60(1 + x)^{60}, \, x \neq 0, \] and \[ (60)^2 S(60) = a(b)^b + b, \] where $a, b \in \mathbb{N}$, then $(a + b)$ is equal to ________.
- Let \( \alpha \beta \gamma = 45 \); \( \alpha, \beta, \gamma \in \mathbb{R} \). If \( x(\alpha, 1, 2) + y(1, \beta, 2) + z(2, 3, \gamma) = (0, 0, 0) \) for some \( x, y, z \in \mathbb{R} \), \( xyz \neq 0 \), then \( 6\alpha + 4\beta + \gamma \) is equal to ______.
View More Questions
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 "|".