Question

which of these relations are funtions ?

Answer 1

I don't see the list of relations you mean — could you paste them?

Meanwhile, here’s how to tell whether a relation is a function:

- If the relation is a set of ordered pairs (a,b), it is a function from domain A to codomain B exactly when each element of the domain appears in at least one pair and appears with exactly one second coordinate. In short: every input has one and only one output.
- If the domain is not explicitly given, assume the domain is the set of all first coordinates appearing in the relation; then the relation is a function iff no first coordinate appears in two different pairs with different second coordinates.
- For a graph in the plane, use the vertical line test: no vertical line should intersect the graph more than once.
- For arrow/diagram representations: each element in the domain must have exactly one arrow leaving it.

Examples:
- {(1,2),(2,3),(3,4)} — function.
- {(1,2),(1,3),(2,4)} — not a function (1 maps to 2 and 3).
- {(1,2),(2,3)} — function if domain = {1,2}; if domain was specified as {1,2,3} then not a function because 3 has no image.

If you paste your relations I’ll mark which are functions and explain why.