Class JEE Mathematics Sets, Relations, and Functions Q #1060
KNOWLEDGE BASED
APPLY
4 Marks 2019 JEE Main 2019 (Online) 12th April Evening Slot MCQ SINGLE
Let A, B and C be sets such that $\phi \neq A \cap B \subseteq C$. Then which of the following statements is not true ?
(A) If (A – B) $\subseteq$ C, then A $\subseteq$ C
(B) B $\cap$ C $\neq$ $\phi$
(C) (C $\cup$ A) $\cap$ (C $\cup$ B) = C
(D) If (A – C) $\subseteq$ B, then A $\subseteq$ B
Prev Next
Correct Answer: D
Explanation
The question provides that $A \cap B \subseteq C$ and $A \cap B \neq \phi$. Analyzing each statement:

Statement A: If $(A - B) \subseteq C$, then $A \subseteq C$. This statement can be verified using Venn diagrams. If the part of A that is not in B is contained in C, then A is a subset of C.

Statement B: $B \cap C \neq \phi$. This means that the intersection of B and C is not empty. Given that $A\cap B \subseteq C$, this is true.

Statement C: $(C \cup A) \cap (C \cup B) = C$. Using the distributive property of sets, $(C \cup A) \cap (C \cup B) = C \cup (A \cap B)$. Since $A \cap B \subseteq C$, then $C \cup (A \cap B) = C$. So, this statement is true.

Statement D: If $(A - C) \subseteq B$, then $A \subseteq B$. This statement is NOT necessarily true. It is possible to have $A - C \subseteq B$, $A \cap B \subseteq C$ and $A \cap B \neq \phi$, but it does NOT require that $A \subseteq B$. This is because A can contain elements outside of B that are also outside of C.

Therefore, the statement that is not true is D.

More from this Chapter

NUMERICAL
In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to ___________.
MCQ_SINGLE
Two newspapers A and B are published in a city. It is known that $25$% of the city populations reads A and $20$% reads B while $8$% reads both A and B. Further, $30$% of those who read A but not B look into advertisements and $40$% of those who read B but not A also look into advertisements, while $50$% of those who read both A and B look into advertisements. Then the percentage of the population who look into advertisement is :-
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 $A = {2, 3, 4, 5, ....., 30}$ and '$\simeq$' be an equivalence relation on $A \times A$, defined by $(a, b) \simeq (c, d)$, if and only if $ad = bc$. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $(4, 3)$ is equal to :
MCQ_SINGLE
Let $\bigcup_{i=1}^{50} X_i = \bigcup_{i=1}^{n} Y_i = T$ where each $X_i$ contains $10$ elements and each $Y_i$ contains $5$ elements. If each element of the set $T$ is an element of exactly $20$ of sets $X_i$'s and exactly $6$ of sets $Y_i$'s, then $n$ is equal to:
View All Questions