Embibe Experts Solutions for Exercise 3: Assignment

Two finite sets have $p$ and $q$ elements respectively. The total number of subsets of first set is $224$ more than the total number of subsets of the second set. Find the values of $p$ and $q$.

For any sets, $A, B, C$ prove that $A-\left(B-C\right)=\left(A-B\right)\cup \left(A\cap C\right)$.

A market research group conducted a survey of $3000$ consumers and reported that $1860$ consumers liked product $A$ and $1950$ consumers liked product $B$. What is the least number of consumers that must have liked both the products?

$A, B, C$ are three sets and $U$ is the universal set. Draw the Venn diagram to represent the following:

(i) ${\left(A-B\right)}^{\text{'}}$

(ii) $A\cup \left(B\cap C\right)$

(iii) ${\left(A\cup B\right)}^{\text{'}}$

If $A=$ set of letters in the word RAJASTHAN and $B=$ set of letters in the word AMRITSAR.
Verify the following results.

$n\left(A-B\right)=n\left(A\right)-n\left(A\cap B\right)=n\left(A\cup B\right)-n\left(B\right)$

Let $A$ and $B$ be two sets such that $n\left(A-B\right)=60+3 x, n\left(B-A\right)=8 x$ and $n\left(A\cap B\right)=x-4$. Draw a venn diagram to illustrate this information. If $n\left(A\right)=n\left(B\right)$, then find

(i) The value of $x$

(ii) $n\left(A\cup B\right)$

If $X$ and $Y$ are subsets of the universal set $U$, then show that

(i) $Y\subset X\cup Y$

(ii) $X\cap Y\subset X$

(iii) $X\subset Y\Rightarrow X\cap Y=X$

Draw the venn diagram to illustrate the following relationship among $A, M, F$ where $A$ is the set of students liking apple, $M$ is the set of students liking mango, $F$ is the set of all the students.

(i) All the students who like mango also like apple, but some students who like apple, do not like mango.

(ii) There is no student who like both mango and apple

(iii) Some of the students like mango but do not like apple, some like apple but do not like mango, and some like both.

(iv) Not all students like mango but every students liking apple also likes mango.