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Consider the following for any three non-empty sets $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$.
1.$A-(B \cup C)=(A-B) \cup(A-C)$
2.$2 \quad A-B=A-(A \cap B)$
3.$\mathrm{A}=(\mathrm{A} \cap \mathrm{B}) \cup(\mathrm{A}-\mathrm{B})$
Which of the above is/are correct?
Options:
1.$A-(B \cup C)=(A-B) \cup(A-C)$
2.$2 \quad A-B=A-(A \cap B)$
3.$\mathrm{A}=(\mathrm{A} \cap \mathrm{B}) \cup(\mathrm{A}-\mathrm{B})$
Which of the above is/are correct?
Solution:
2590 Upvotes
Verified Answer
The correct answer is:
2 and 3
A Venndiagram ofthe three non-empty andintersecting sets is drawn, dividing into 7 regions, a to gas showr below and consider statements one by one Statement $(1): \mathrm{A}-(\mathrm{B} \cup \mathrm{C})=(\mathrm{A}-\mathrm{B}) \cup(\mathrm{A}-\mathrm{C})$
LHS = regions $a, b, c, d-b, c, d, e, f, g=a$.
$\mathrm{RHS}=\mathrm{a}, \mathrm{d}, \cup \mathrm{a}, \mathrm{b}=\mathrm{a}, \mathrm{b}, \mathrm{d}$
So, statement $(1)$ is not correct. Statement $(2): A-B=A-(A \cap B)$
LHS=region a, $\mathrm{d}$. $\mathrm{RHS}=$ region $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}-($ region $\mathrm{b}, \mathrm{c},=\mathrm{a}, \mathrm{d}$
So, statement $(2)$ iscorrect. Statement $(3): A=(A \cap B) \cup(A-B)$
LHS = regions $a, b, c, d$. $\mathrm{RHS}=$ regions $\mathrm{b}, \mathrm{c}, \cup$ region $\mathrm{a}, \mathrm{d}=\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ so,
statement (3) is correct.
LHS = regions $a, b, c, d-b, c, d, e, f, g=a$.
$\mathrm{RHS}=\mathrm{a}, \mathrm{d}, \cup \mathrm{a}, \mathrm{b}=\mathrm{a}, \mathrm{b}, \mathrm{d}$
So, statement $(1)$ is not correct. Statement $(2): A-B=A-(A \cap B)$
LHS=region a, $\mathrm{d}$. $\mathrm{RHS}=$ region $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}-($ region $\mathrm{b}, \mathrm{c},=\mathrm{a}, \mathrm{d}$
So, statement $(2)$ iscorrect. Statement $(3): A=(A \cap B) \cup(A-B)$
LHS = regions $a, b, c, d$. $\mathrm{RHS}=$ regions $\mathrm{b}, \mathrm{c}, \cup$ region $\mathrm{a}, \mathrm{d}=\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ so,
statement (3) is correct.
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