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Question
Let\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = \text{B and C} = \left\{ x \in R : x \geq 0 \right\} and\]\[S = \left\{ \left( x, y \right) \in A \times B : x^2 + y^2 = 1 \right\} \text{and } S_0 = \left\{ \left( x, y \right) \in A \times C : x^2 + y^2 = 1 \right\}\]
Then,
Options
S defines a function from A to B
`S_0` defines a function from A to C
S0 defines a function from A to B
S defines a function from A to C
Solution
(a) S defines a function from A to B
\[\text{Let x} \in A\]
\[ \Rightarrow - 1 \leq x \leq 1\]
\[\text{Now}, x^2 + y^2 = 1\]
\[ \Rightarrow y^2 = 1 - x^2 \]
\[ \Rightarrow y = \pm \sqrt{1 - x^2}\]
\[ \Rightarrow - 1 \leq y \leq 1\]
\[ \therefore y \in B\]
\[\text{Thus, S defines a function from A to B} . \]
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