Subjective Type

If $$U=\left\{1,2,3,4,5,6,7,8,9\right\}, A=\left\{2,4,6,8\right\}$$ and $$B=\left\{2,3,5,7\right\}$$, verify that: $$(A\cup B)'=(A'\cap B')$$

Solution

Given $$U=\left\{ 1,2,3,4,5,6,7,8,9 \right\} $$
$$A=\left\{ 2,4,6,8 \right\} $$, $$B=\left\{ 2,3,5,7 \right\} $$

1) $$A\cup B=\left\{ 2,4,6,8 \right\}\cup \left\{ 2,3,5,7 \right\}$$
$$\therefore A\cup B=\left\{ 2,3,4,5,6,7,8 \right\} $$

$$\therefore { \left( A\cup B \right) }^{ \prime }=\left\{ x:x\notin \left( A\cup B \right) \quad and\quad x\in U \right\} $$
$$\therefore { \left( A\cup B \right) }^{ \prime }=\left\{ 1,9 \right\} $$ Equation (1)

2) $${ A }^{ \prime }=\left\{ x:x\notin A\quad and\quad x\in U \right\} $$
$$\therefore { A }^{ \prime }=\left\{ 1,3,5,7,9 \right\} $$

$${ B }^{ \prime }=\left\{ x:x\notin B\quad and\quad x\in U \right\} $$
$$\therefore { B }^{ \prime }=\left\{ 1,4,6,8,9 \right\} $$

$${ A }^{ \prime }\cap { B }^{ \prime }=\left\{ 1,3,5,7,9 \right\}\cap \left\{ 1,4,6,8,9 \right\}$$
$$\therefore { A }^{ \prime }\cap { B }^{ \prime }=\left\{ 1,9 \right\} $$ Equation (2)

From equation (1) and (2),
$${ \left( A\cup B \right) }^{ \prime }={ A }^{ \prime }\cap { B }^{ \prime }$$

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