Let `A={x in R :-1lt=xlt=1}=B` and `C={x in R : xgeq0}` and let `S={(x ,\ y) in AxxB : x^2+y^2=1}` and `S_0={(x ,\ y) in AxxC : x^2+y^2=1}dot` Then `S` defines a function from `A` to `B` (b) `S_0` defines a function from `A` to `C` (c) `S_0` defines a function from `A` to `B` (d) `S` defines a function from `A` to `C`

Options

A. S defines a function from A to B
B. `S_(0)` defines a function from A to C
C. `S_(0)` defines a function from A to b
D. S defines a function from A to c

Answer

B

Explanation

We observe that S denotes the set of all points on the circle `x^(2)+y^(2)=1 and S_(0)` denotes the set of all points on the semi-circle `x^(2)+y^(2)=1` lying above x-axis. Clearly ,`(1//2, sqrt3//2),(1//2,-sqrt3//2) in S`. Infact corresponding to every value of `x in A` there are two values of `y in B` Therefore, S is not a function from A to B. Clearly `S_(0) subset AxxC` such that for each `x in A` there is unique value of `y in C`. Therefore, `S_(0)` is a function from A to C.
ALTER
If a straight line parallel to y-axis cuts the graph of a curve at more than one-point them the set of all points on it does not define a function. However, the set of all points on a curve defines a function when no line parallel to y-axis crosses it more than once.