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Divisibility rules.

Odd numbers are never 0.[1]

Contents

[edit] Sets

Subobject classifier

[edit] Spaces

There are three non-isomorphic ways to make a two-element set into a topological space: the trivial topology, the particular point topology, and the discrete topology.

[edit] Trivial space

[edit] Sierpiński space

Main article: Sierpiński space

[edit] 0-sphere

The only one of these that is T1 is the two-element set with the discrete topology, which may simply be called 2. The discrete two-element space is the prototype for disconnected spaces; a space X is disconnected if and only if there exists a continuous surjection from X to 2. Individual maps from 2 to X are less informative, since they are always continuous, but homotopy classes of such maps again measure the disconnectedness of X, using the methods of algebraic topology.

Since {1, -1} is the set of unit vectors in R1, it may be identified as S0, the zero-dimensional sphere. S0 is a pointed space with basepoint 1, and it has nice properties with respect to some common topological constructions:

Positive-dimensional spheres can be expressed as repeated suspensions of S0.

Unlike every positive sphere, S0 itself is not the suspension of anything, and there is no way to make it a cogroup object in the pointed homotopy category hTop•. Concretely, there cannot exist a map from S0 to the wedge sum S0 V S0 with the property that collapsing either of the two addends makes it homotopic to the identity. As a consequence, the "zeroth homotopy group" π0(X) does not inherit a natural group structure from S0. It can still be a group if X itself is a topological group; then π0(X) is the group of components of X. Otherwise, π0(X) is merely the pointed set of components of X.

[edit] Algebras

[edit] Cyclic group of order 2

Finding a homomorphism from 2 to G is the same as finding an element of G of order 2; finding a homomorphism from G to 2 is the same a subgroup of G of index 2. The direct product of 2 with itself is the Klein four-group; the free product of 2 with itself is the infinite dihedral group.

2 is the only nontrivial group with a trivial automorphism group.[2]

The classifying space of 2 is infinite-dimensional real projective space, for which the total space is the infinite-dimensional sphere.

[edit] Field

[edit] Boolean algebra

Main article: Two-element Boolean algebra

[edit] References

  1. ^ Wagstaff, Samuel S. (2003). Cryptanalysis of Number Theoretic Ciphers. CRC Press, 23. ISBN 1584881534. 
  2. ^ Hedrlín, Zdenek and Joachim Lambek (February 1969). "How comprehensive is the category of semigroups?". Journal of Algebra 11 (2): pp. 195-212. doi:10.1016/0021-8693(69)90054-4.  See p. 196.

Two rational numbers can be added together to get another rational number. In practice, rational addition is defined and carried out in terms of fractions:

\frac ab + \frac cd = \frac{ad+bc}{bd}

It is straightforward to prove that the above operation is well-defined as an operation on rational numbers, in that it respects equivalence of fractions. It is compatible with the simpler addition operation on the integers. Rational addition also possesses the usual properties expected of a well-behaved addition operation: associativity, commutativity, and the existence of an additive identity and additive inverses.

The concrete task of adding together two fractions is notoriously difficult to teach when it arises in primary education.

[edit] References