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
The number of elements in the set $\{ n \in Z:|{n^2} - 10n + 19| < 6\} $ is _________.
MCQ_SINGLE
Consider the relations $R_1$ and $R_2$ defined as $aR_1b \Leftrightarrow a^2 + b^2 = 1$ for all $a, b \in R$ and $(a, b)R_2(c, d) \Leftrightarrow a+ d = b + c$ for all $(a, b), (c, d) \in N \times N$. Then:
NUMERICAL
Let A = {n $\in$ N | n2 $\le$ n + 10,000}, B = {3k + 1 | k$\in$ N} an dC = {2k | k$\in$N}, then the sum of all the elements of the set A $\cap$(B $-$ C) is equal to _____________.
MCQ_SINGLE
Let $A = {1, 3, 4, 6, 9}$ and $B = {2, 4, 5, 8, 10}$. Let $R$ be a relation defined on $A \times B$ such that $R = {((a_1, b_1), (a_2, b_2)): a_1 \le b_2 \text{ and } b_1 \le a_2}$. Then the number of elements in the set R is :
MCQ_SINGLE
An organization awarded $48$ medals in event 'A', $25$ in event 'B' and $18$ in event 'C'. If these medals went to total $60$ men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?
View All Questions