Imagine a school dance. There is a set of boys and a set of girls. When the music starts, each boy picks a girl to dance with.
Think of this as a mapping from a boy to a girl, or from an element in the set of boys to an element in the set of girls.
The mapping is said to be injective (or one-to-one) if each boy picks a different girl. If two boys try to dance with the same girl, the mapping isn't injective.
The mapping is said to be surjective (or onto) if no girls are left without a partner. If there is a girl not dancing, the mapping isn't surjective.
If the mapping is both injective and surjective it is said to be bijective.
You can immediately tell if there are the same number of boys and girls if the mapping is bijective—in other words, each boy is dancing with one and only one girl and no girls are left without a boy to dance with.
The existence of a bijection can be used to demonstrate that two sets have same number of elements or, in the case of infinite sets, have the same cardinality.
Bijections are also very important in establishing the equivalence between two structured sets (for example between two topological spaces) as we shall see in the near future.
UPDATE: next post
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.