Class JEE Mathematics Sets, Relations, and Functions Q #1073
KNOWLEDGE BASED
APPLY
4 Marks 2009 AIEEE MCQ SINGLE
If $A$, $B$ and $C$ are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$, then :
(A) $A = C$
(B) $B = C$
(C) $A \cap B = \phi$
(D) $A = B$
Prev Next
Correct Answer: B
Explanation
Given $A \cap B = A \cap C$ and $A \cup B = A \cup C$. From the principle of inclusion-exclusion: $|A \cup B| = |A| + |B| - |A \cap B|$ and $|A \cup C| = |A| + |C| - |A \cap C|$. Since $A \cup B = A \cup C$ and $A \cap B = A \cap C$, then $|A \cup B| = |A \cup C|$ and $|A \cap B| = |A \cap C|$. Thus, $|A| + |B| - |A \cap B| = |A| + |C| - |A \cap C|$. This simplifies to $|B| = |C|$. Since $A \cap B = A \cap C$ and $A \cup B = A \cup C$, it means that $B$ and $C$ contain the same elements and therefore are equal to each other ($B=C$).

More from this Chapter

NUMERICAL
Let $S=\left\{p_1, p_2 \ldots, p_{10}\right\}$ be the set of first ten prime numbers. Let $A=S \cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs $(x, y), x \in S$, $y \in A$, such that $x$ divides $y$, is ________ .
MCQ_SINGLE
Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) ∈ R if and only if max{x, y} ∈ {3, 4}. Then among the statements (S1): The number of elements in R is 18, and (S2): The relation R is symmetric but neither reflexive nor transitive
NUMERICAL
Let $A = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\min \,\{ i,j\} } } $ and $B = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\max \,\{ i,j\} } } $. Then A + B is equal to _____________.
MCQ_SINGLE
Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :
NUMERICAL
Let $A=\{1,2,3,4,5,6,7\}$. Define $B=\{T \subseteq A$ : either $1 \notin T$ or $2 \in T\}$ and $C=\{T \subseteq A: T$ the sum of all the elements of $T$ is a prime number $\}$. Then the number of elements in the set $B \cup C$ is ________________.
View All Questions