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Relation — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSRelation3 Marks
Question
If A = {1, 2, 3}, B = {4}, C = {5}, then verify that: $\text{A}\times(\text{B}\cap\text{C})=(\text{A}\times\text{B})\cap(\text{A}\times\text{C})$
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Answer

We have, $\text{A}=\{1,2,3\},\text{ B}=\{4\}$ and $\text{C}=\{5\}$ $\therefore\ \text{B}\cap\text{C}=\{4\}\cap\{5\}=\phi$ $\therefore\ \text{A}\times(\text{B}\cap\text{C})=\{1,2,3\}\times\phi$ $\Rightarrow\text{A}\times(\text{B}\cap\text{C})=\phi\ ...(\text{i})$ Now, $\text{A}\times\text{B}=\{1, 2, 3\}\times\{4\}$ $=\{(1, 4), (2, 4), (3, 4)\}$ and, $\text{A}\times\text{C}=\{1,2,3\}\times\{5\}$ $=\{(1, 5) , (2, 5), (3, 5)\}$ $\therefore\ (\text{A}\times\text{B})\cap(\text{A}\times\text{C})=\{(1, 4), (2, 4), (3, 4)\} \cup\{(1, 5), (2, 5), (3, 5)\}$ $\Rightarrow(\text{A}\times\text{B})\cup(\text{A}\times\text{C})=\phi\ ...(\text{ii})$ From equation (i) and (ii), we get $\text{A}\times (\text{B}\cap\text{C})=(\text{A}\times\text{B})\cap(\text{A}\times\text{C})$ Hence verified.

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