JEE MAIN - Mathematics (2019 - 12th April Evening Slot - No. 1)
Let A, B and C be sets such that $$\phi $$ $$ \ne $$ A $$ \cap $$ B $$ \subseteq $$ C. Then which of the following statements is not true ?
If (A – B) $$ \subseteq $$ C, then A $$ \subseteq $$ C
B $$ \cap $$ C $$ \ne $$ $$\phi $$
(C $$ \cup $$ A) $$ \cap $$ (C $$ \cup $$ B) = C
If (A – C) $$ \subseteq $$ B, then A $$ \subseteq $$ B
Explanation
According to the question, we have the following Venn diagram.
Here, $A \cap B \subseteq C$ and $A \cap B \neq \phi$
_12th_April_Evening_Slot_en_1_1.png)
Now, from the Venn diagram, it is clear that $B \cap C \neq \phi$, is true
Also, $(C \cup A) \cap(C \cup B)=C \cup(A \cap B)=C$ is true.
If $(A-B) \subseteq C$, for this statement the Venn diagram is
_12th_April_Evening_Slot_en_1_2.png)
From the Venn diagram, it is clear that if $A-B \subseteq C$, then $A \subseteq C$.
Now, if $(A-C) \subseteq B$, for this statement the Venn diagram.
_12th_April_Evening_Slot_en_1_3.png)
From the Venn diagram, it is clear that
$A \cap B \neq \phi, A \cap B \subseteq C$ and $A-C=\phi \subseteq B$ but $A \subseteq B$
Here, $A \cap B \subseteq C$ and $A \cap B \neq \phi$
Now, from the Venn diagram, it is clear that $B \cap C \neq \phi$, is true
Also, $(C \cup A) \cap(C \cup B)=C \cup(A \cap B)=C$ is true.
If $(A-B) \subseteq C$, for this statement the Venn diagram is
From the Venn diagram, it is clear that if $A-B \subseteq C$, then $A \subseteq C$.
Now, if $(A-C) \subseteq B$, for this statement the Venn diagram.
From the Venn diagram, it is clear that
$A \cap B \neq \phi, A \cap B \subseteq C$ and $A-C=\phi \subseteq B$ but $A \subseteq B$
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