From the tally symbols for numbers m and n we can form another tally symbol by putting
all the tally marks within the same space, and this combined tally symbol will also
represent a number. This process is known as **addition**. Thus, provided we can carry
out this process, we can **add** two numbers m and n to form a number called their
**sum** which is written using the **plus** sign (+) in the form m+n. For example
(||)+(|||) = (|||||), i.e. 2+3 = 5.

Addition table (base ten)

+ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

0 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |

2 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

3 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |

4 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |

5 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

6 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

7 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |

8 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |

9 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |

10 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

Any number can be represented uniquely as a sum of a sequence of ones.
This statement can be expressed as n = ∑(1,1,1,...), where ∑ is the **summation operator**,
and is true of 0 and 1 as well as of higher numbers, since 0 = ∑( ), sum of an empty array of units,
and 1 = ∑(1), sum of an array of a single unit.

Addition is commutative: a+b = b+a, i.e. the result is independent of the order of the numbers. Addition is associative: a+(b+c) = (a+b)+c, i.e. the result of a series of additions is independent of the order in which they are carried out. The number 0, has the additive identity property: a+0 = a for any number a. In particular 0+0 = 0.

Order and addition are related: we have n>m if and only if there exists a number d, not zero, such that n = m+d. The number n+1 is the successor of n.

If n = m+d then we say that the **difference** of n and m is d.
This number is uniquely determinate and so we are justified in introducing another operation symbol,
writing n−m = d and we call this operation **subtraction**.
We have m = n if and only if n−m = 0.
Subtraction is only a partial operation on numbers, since n−m is defined only when n≥m.

Order and subtraction are related: we have n<m if and only if there exists a nonzero number d such that m−d = n. The number n−1 is the immediate predecessor of n. The number of spaces between n things in a row is n−1. This simple principle often turns up. (On the other hand, the number of spaces between n things arranged in a circle is n.)

We can also define the **symmetric difference** nHm to be the number d such that
either n = m+d or m = n+d, one of which must be true (both are true only when n=m, then d=0).

Addition is defined by combining separate sets. More generally two sets will have a non-null intersection, and in this case the numbers of elements in the union and intersection are related to the numbers in the given sets by: |A∪B| + |A∩B| = |A| + |B|.

The number of elements in the union of two overlapping sets is given by
|AuB| = (|A| + |B|) − |A∩B|. This result can be generalised to apply to a family of sets:
|∪{A,B,C,D,...}| = (∑|A| + ∑|N{A,B,C}| + ∑|N{A,B,C,D,E}| + ...) – (∑|N{A,B}| + ∑|N{A,B,C,D}| + ...)
that is the difference between the sums of the numbers of elements common to odd and even numbers of the sets.
This rule is known as the **principle of inclusion and exclusion** (or of 'cross-classification').

Example: Out of a set of 120 students, 42 take English, 37 take neither language, 9 take both languages,
how many take French but not English?

Answer; |E∪F| = 120−37 = 83, |E| = 42, |F−E| = |E∪F|−|E| = 83−42 = 41.
The information about |F∩E| is not needed.