Examples
1. f: {1,2,3} → {a,b,c,d} defined by
The image of the set {2,3} under f is f({2,3}) = {a,c}. The image of the function f is {a,c}. The preimage of a is f −1({a}) = {1,2}. The preimage of {a,b} is also {1,2}. The preimage of {b,d} is the empty set {}.
2. f: R → R defined by f(x) = x2.
The image of {-2,3} under f is f({-2,3}) = {4,9}, and the image of f is R+. The preimage of {4,9} under f is f −1({4,9}) = {-3,-2,2,3}. The preimage of set N = {n ∈ R | n < 0} under f is the empty set, because the negative numbers do not have square roots in the set of reals.
3. f: R2 → R defined by f(x, y) = x2 + y2.
The fibres f −1({a}) are concentric circles about the origin, the origin itself, and the empty set, depending on whether a>0, a=0, or a<0, respectively.
4. If M is a manifold and π :TM→M is the canonical projection from the tangent bundle TM to M, then the fibres of π are the tangent spaces Tx(M) for x∈M. This is also an example of a fiber bundle.
Read more about this topic: Image (mathematics)
Famous quotes containing the word examples:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)