Question 15 Marks

If $A =\{p, q, r, s\}, B =\{q, s, u\}$ and $C =\{r, s, t, u\}$, then prove the following :
(i) $(A-B) \cup(A-C)=A-(B \cap C)$
(ii) $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$
(iii) $A \cup(B \cap C)=(A \cup B) \cap(A \cup C)$

Answer

$(i)$
$\begin{aligned}A-B & =\{p, q, r, s\}-\{q, s, u\} \\& =\{p, r\} \\\text { and } \quad A-C & =\{p, q, r, s\}-\{r, s, t, u\} \\& =\{p, q\} \\\therefore \quad(A-B) \cup(A-C) & =\{p, r\} \cup\{p, q\} \\& =\{q, r, p\}\ldots\ldots\text {(i)} \\\text { and } \quad & B \cap C  =\{q, s, u\} \cap\{r, s, t, u\} \\& =\{s, u\} \\\therefore & A-(B \cap C)  =\{p, q, r, s\}-\{s, u\} \\& =\{p, q, r\}\ldots\ldots \text {(ii)}\end{aligned}$
From equations (i) and (ii)
$(A-B) \cup(A-C)=A-(B \cap C)$
Hence proved.
$(ii)$
$\begin{aligned}B \cup C & =\{q, s, u\} \cup\{r, s, t, u\} \\& =\{q, r, s, t, u\} \\\therefore \quad A \cap(B \cup C) & =\{p, q, r, s\} \cap\{q, r, s, t, u\} \\& =\{q, r, s\}\ldots\ldots\text {(i)} \\A \cap B & =\{p, q, r, s\} \cap\{q, s, u\} \\& =\{q, s\} \\A \cap C & =\{p, q, r, s\} \cap\{r, s, t, u\} \\& =\{r, s\} \\\therefore \quad(A \cap B) \cup(A \cap C) & =\{q, s\} \cup\{r, s\} \\& =\{q, r, s\}\ldots\ldots\text {(ii)}\end{aligned}$
From equations (i) and (ii)
$A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$
Hence proved.
$(iii)$
$\begin{aligned}B \cap C & =\{q, s, u\} \cap\{r, s, t, u\} \\& =\{s, u\} \\A \cup(B \cap C) & =\{p, q, r, s\} \cup\{s, u\} \\
& =\{p, q, r, s, u\} \ldots \ldots\text {(i)} \$A \cup B) & =\{p, q, r, s\} \cup\{q, s, u\} \\& =\{p, q, r, s, u\} \\A \cup C & =\{p, q, r, s\} \cup\{r, s, t, u\} \\& =\{p, q, r, s, t, u\} \\\therefore \quad(A \cup B) \cap(A \cup C) & =\{p, q, r, s, u\} \cap \\& \{p, q, r, s, t, u\} \\& =\{p, q, r, s, u\}\ldots\ldots\text {(ii)}\end{aligned}$
$\therefore$ From equations (i) and (ii)
$A \cup(B \cap C)=(A \cup B) \cap(A \cup C)$
Hence proved.

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