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Sets — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsSets1 Mark
MCQ
For any three sets A, B and C:
  • ✓
    $\text{A}\cap\text{(B} -\text{C)}=\text{(A}\cap\text{B)} - \text{(A}\cap\text{C)}$
  • B
    $\text{A}\cap\text{(B} -\text{C)}=\text{(A}\cap\text{B)}- \text{C}$
  • C
    $\text{A}\cup\text{(B} - \text{C)}=\text{(A}\cup\text{B)}\cap\text{(A}\cup\text{C}')$
  • D
    $\text{A}\cup\text{(B} - \text{C)}=\text{(A}\cup\text{B)}-\text{(A}\cup\text{C}).$
✓

Answer

Correct option: A.
$\text{A}\cap\text{(B} -\text{C)}=\text{(A}\cap\text{B)} - \text{(A}\cap\text{C)}$
Let x be any arbitrary element of $\text{A}\cap\text{B}-\text{C.}$
Thus, we have,
$\text{x}\in\text{A}\cap\text{(B - C)}\Rightarrow\text{x}\in\text{A}$ and $\text{x}\in\text{B}-\text{C}$
$\Rightarrow\text{x}\in\text{A}$ and $\text{(x}\in\text{B and x}\not\in\text{C)}$
$\Rightarrow\text{x}\in\text{A and x}\in\text{B}$ and $\Rightarrow\text{X}\in\text{A and x}\not\in\text{C}$
$\Rightarrow\text{x(A}\cap\text{B)}$ and $\text{x}\not\in\text{(A}\cap\text{C)}$
$\Rightarrow\text{x}\in[\text{(A}\cap\text{B)}-\text{(A}\cap\text{C)}]$
$\Rightarrow\text{A}\cap\text{(B}-\text{C)}\subseteq\text{(A}\cap\text{B)} - \text{(A}\cap\text{C)}$
Similarly, $\text{(A}\cap\text{B)}-\text{(A} - \text{C)}\subseteq\text{(A}\cap\text{(B}-\text{C)}$
Hence, $\text{A}\cap\text{(B} - \text{C)}=\text{(A}\cap\text{B)} - \text{(A}\cap\text{C)}.$

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