Sets - Cardinal Numbers of Two Finite Sets-advanced

In a group of $100$ people, $70$ can speak Hindi and $45$ can speak English. If every person speaks at least one of the two languages, find: (i) how many can speak both Hindi and English? (ii) how many can speak Hindi only?

Let $H$ be the set of people who speak Hindi and $E$ be the set of people who speak English. Given: $n(H \cup E) = 100$ $n(H) = 70$ $n(E) = 45$

(i) Using the formula: $n(H \cup E) = n(H) + n(E) - n(H \cap E)$ $100 = 70 + 45 - n(H \cap E)$ $100 = 115 - n(H \cap E)$ $n(H \cap E) = 115 - 100 = 15$ So, $15$ people speak both languages.

(ii) Number of people who speak Hindi only: $n(H - E) = n(H) - n(H \cap E)$ $n(H - E) = 70 - 15 = 55$ So, $55$ people speak Hindi only.

We use the addition principle of sets. Since everyone speaks at least one language, the union equals the total group size. Hindi only is calculated by subtracting the overlap (intersection) from the total Hindi speakers.

In a survey of $500$ car owners, $400$ owned car $A$ and $200$ owned car $B$. $50$ owned both $A$ and $B$. Is this data correct?

Let $n(A) = 400$, $n(B) = 200$, and $n(A \cap B) = 50$. We calculate the number of people who own at least one car: $n(A \cup B) = n(A) + n(B) - n(H \cap B)$ $n(A \cup B) = 400 + 200 - 50$ $n(A \cup B) = 550$

However, the total number of people surveyed is given as $500$. Since $n(A \cup B)$ cannot be greater than the universal set $n(U)$, we have: $550 > 500$ This is a contradiction.

The cardinality of the union of subsets can never exceed the cardinality of the universal set. Here, the calculated union ($550$) exceeds the total surveyed ($500$), proving the data is inconsistent.

If $n(U) = 700$, $n(A) = 200$, $n(B) = 300$ and $n(A \cap B) = 100$, find $n(A' \cap B')$.

First, find $n(A \cup B)$: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ $n(A \cup B) = 200 + 300 - 100 = 400$

By De Morgan's Law, $A' \cap B' = (A \cup B)'$. $n(A' \cap B') = n(U) - n(A \cup B)$ $\begin{array}{r} 700 \\ - 400 \\ \hline 300 \end{array}$ So, $n(A' \cap B') = 300$.

We use the relation between the intersection of complements and the complement of the union. After finding the union, we subtract it from the universal set to find the elements outside both $A$ and $B$.