## 16.1 Discrete Mathematics

The mathematical concepts we build on include:

sets
A set has elements (members). We write s ∈S if s is an element of set S. The elements in a set define the set, so that two sets are equal if they have the same elements.
tuples
An n-tuple is an ordered grouping of n elements, written ⟨x1,...,xn. A 2-tuple is a pair, and a 3-tuple is a triple. Two n-tuples are equal if they have the same members in the corresponding positions. If S is a set, Sn is the set of n-tuples ⟨x1,...,xn where xi is a member of S. S1 ×S2×...×Sn is the set of n-tuples ⟨x1,...,xn where each xi is in Si.
relations
A relation is a set of n-tuples. The tuples in the relation are said to be true of the relation.
functions
A function, or mapping, f from set D, the domain of f, into set R, the range of f, written f:D →R, is a subset of D×R such that for every d∈D there is a unique r∈R such that ⟨d,r⟩ ∈f. We write f(d)=r if ⟨d,r⟩∈f.

While these may seem like obscure definitions for common-sense concepts, you can now use the common-sense concepts comfortable in the knowledge that if you are unsure about something, you can check the definitions.