math111_logo Transformation

2. Onto

Let T: XY be a transformation. Recall that T is onto (surjective) if its range is the whole target set. More specifically, this means

For any yY, there is xX, such that T(x) = y.

The following characterizes onto transformations. More properties can be found in this exercise.

A transformation T: XY is onto ⇔ There is a transformation S: YX, such that TS = idY.

Proof If TS = idY, then for any yY, the element x = S(y) satisfies T(x) = TS(y) = idY(y) = y. This proves that any element of Y is in the range.

Conversely, suppose T is onto. Then for any yY, there is xX, such that (x, y) ∈ T. We choose S(y) to be one of such x. After S(y) is chosen for all yY, we have a transformation S: YX. By the way S is constructed, we clearly have TS = idY.

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