Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B. The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3} .
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Anonymous2013-08-31 22:32
Ludwig Wittgenstein questioned the way Zermelo–Fraenkel set theory handled infinities.Wittgenstein's views about the foundations of mathematics were later criticised by Georg Kreisel and Paul Bernays, and investigated by Crispin Wright, among others.
Until the late 19th century, the axiom of choice was often used implicitly, although it had not yet been formally stated. For example, after having established that the set X contains only non-empty sets, a mathematician might have said "let F(s) be one of the members of s for all s in X."
Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset. The restricted principle "Every partially ordered set has a maximal totally ordered subset" is also equivalent to AC over ZF.
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Anonymous2013-09-01 0:48
In all models of ZF¬C there is a vector space with no basis.
automorphism if f is both an endomorphism and an isomorphism. aut(a) denotes the class of automorphisms of a.
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Anonymous2013-09-01 2:18
The morphism f has a left inverse if there is a morphism g:Y → X such that g ∘ f = idX. The left inverse g is also called a retraction of f.[1] Morphisms with left inverses are always monomorphisms, but the converse is not always true in every category; a monomorphism may fail to have a left-inverse.
Tangent and cotangent bundles: The map which sends every differentiable manifold to its tangent bundle and every smooth map to its derivative is a covariant functor from the category of differentiable manifolds to the category of vector bundles. Likewise, the map which sends every differentiable manifold to its cotangent bundle and every smooth map to its pullback is a contravariant functor.