Definition - Q is an operation if and only if the following statements are true:
C is a set.
Q is a set of ordered pairs, P = (s, t).
Each s and each t in each ordered pair, P, is also in C.
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Definition - Q is an binary operation if and only if the following statements are true:
C is a set.
Q is a set of ordered pairs, P = (s, t).
Each s and each t in each ordered pair, P, is also in C.
P is such that s = {a, b} or s = (a, b) where a and b are numbers.
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Definition - X is the symbol for operation Q if and only if X is a single character and X is consistently used to notate Q.
Definition - Q is an expression if and only if Q is a written entity that includes an operation or more than a single operation together with a number or more than a single number and does not include an equal sign.





Definition - The absolute value of integer x = v if and only if the following statements are true:
Q is an integer set containing x.
If x is in the set of whole numbers of Q, then v = x.
if x is in the set of negative integers of Q, then v and x are symmetric.
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The absolute value of x is a positive integer whether x is positive or negative.   If x = 0, then v = 0.

Definition - Integer x and integer y are of the same sign if and only if the following statements are true:
Q is an integer set containing x and y.
Neither x nor y is positive; or, neither x not y is negative.
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Definition - Element x is equal to element y if and only if the following statements are true:
Q is an ordered set of non-set elements containing x and y.
Integer j is the starting element of Q.
R is the string in Q from j to x.
T is the string in Q from j to y.
| R | = | T |.
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Notation - "Element x is equal to element y" may be written this way: x = y

Definition - Element x is greater than element y if and only if the following statements are true:
Q is an ordered set of non-set elements containing x and y.
Integer j is the starting element of Q.
R is the string in Q from j to x.
T is the string in Q from j to y.
| R | > | T |.
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Notation - "Element x is greater than element y" may be written this way: x > y

Definition - Element x is less than element y if and only if the following statements are true:
Q is an ordered set of non-set elements containing x and y.
Integer j is the starting element of Q.
R is the string in Q from j to x.
T is the string in Q from j to y.
| R | < | T |.
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Notation - "Element x is less than element y" may be written this way: x < y

Recall that N is a number if and only if N is an element in ordered set S, and each element in S is not a set. This means that the previous three definitions define equal to, greater than and less than for numbers, whether they are integers in an integer set or elements in any other ordered set of non-set elements.

Definition - The sum of integer x and integer y equals integer z if and only if the following statements are true:
Q is an integer set containing x and y.
Integer j is the starting element of Q, and k is the ending element of Q.
Integer v is the absolute value of x, and w is the absolute value of y.
The size of the string in Q from w to k is greater than the size of the string in Q from 0 to w.
The size of the string in Q from v to k is greater than the size of the string in Q from 0 to v.
If y is not negative, the string in Q from x to z has a one-to-one correspondence with the string in Q from 0 to y.
If y is not positive, the string in Q from z to x has a one-to-one correspondence with the string in Q from y to 0.
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The first statement and last two statements in the definition of the sum would suffice if the size of the integer set could be known to be such that the sum would assuredly be in the integer set.   The other statements assure that the sum is defined only for such an integer set.

It can be inferred from the definition of the the term integer set that the size of the string in Q from j to 0 is equal to the size of the sting in Q from 0 to k because an integer set is comprised of a center element together with a pair of ordered sets that have a one-to-one correspondence.

Definition - Integer z equals integer x minus integer y if and only if z is the sum of x and w where w is the negative of y.
Definition - Integer z equals the difference of integer x and integer y if and only if z equals x minus y.
Definition - Q is integer addition if and only if for some integer set containing x and y, Q is the sum of x and y.
Definition - Q is integer subtraction if and only if for some integer set containing x and y, Q is the difference of x and y.

Notice that, in such a case, Q is an operation and Q is a binary operation.

Notation - The sum of x and y may be written x + y. The "+" symbol must appear between x and y in the expression. In this case, the plus sign is the symbol for integer addition.

Notation - The difference of x and y may be written x - y. The "-" symbol must appear between x and y in the expression. In this case, the minus sign is the symbol for integer subtraction.

Notice that if integer x = y then x - y = 0.

The sum of multiple integers such as 1 + w + x + y + z + 0 + p + q can be discovered through a process that goes through time. In considering the sum of a set of integers containing more than a pair of integers, each integer is added to the current sum exactly once until all of the integers in the set have been added to the sum. The sum is initially equal to 0. Each time the sum is added to, the sum changes. Precisely defining such a process requires definitions that use the terms time, before, after, first, subsequently, after, occur, happen and the like. These will be used as undefined terms.

Definition - P is a path set if and only if the following statements are true:
P is a set containing at least a pair of elements.
Element x is in P if and only if x is an event or a condition.
One and only one element j in P is such that no other element in P is associated with it.
One and only one element k in P is not associated with any element in P.
Each element x in Ps other than element k in P is associated with some other element y in P, or upon conditions stated in the definition of x is instead associated with one of a list of alternative elements in P.
Element x is associated one and only one alternative element at any given time.
Element x is not associated with any element other than those listed.
No listed alternative element is the element k in P such that k is not associated with any element in P.
No listed alternative element is the element j in P such that no other element in P is associated with it.
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Notation - P = (t, u, v, w, [x :: u, y], y, z)

P is a set in which a succession of events or conditions progress through time until x occurs. Depending upon conditions stated in the definition of x, the next event or condition to occur is either u or y. The square brackets enclose the name of the element that makes a choice together with the names of the choices written immediately to the right of the double colon.

Successions of events having more than one alternative element per choice are written this way:

A = (q, r, s, t, u, v, w, [x :: r, t, u, y], y, z)        The definition of x calls for four possible next events or conditions (elements).

Successions of events having more than one element making a choice are written this way:

B = (q, r, s, [t :: r, w] w, [x :: r, u, y], y, z)        Both t and x are actions specify calculate a choice.

Definition - Element e is removed from set K if and only if the following statements are true:
E is a path set.
E = (A, B, C)
A = K is a non-empty unordered set containing element e; and X is an unordered set; and y is in X if and only if y not equal to e, and y is in K.
B = K becomes undefined and discarded.
C = X is renamed to be known as K.
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Definition - Integer z is the sum of all of the integers in K if and only if the following statements are true:
E is a path set.
E = (A, B, [C :: B, D], D)
A = Q is an integer set, and z in Q equals 0 in Q, and K is an unordered set; and K is either empty, or K is a subset of Q, or K is coincident with Q.
B = If x is in K, then z becomes the sum of x and the previous value of z, and x is removed from K.
C = If K is not empty, repeat B; otherwise continue to D.
D = Stop.
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Definition - The product of integer x and integer y equals integer z if and only if the following statements are true:
Q is an integer set containing x, y and z.
S is a set.
Each element in S is a set.
The intersection of all of the sets in S is empty.
If set r is in S, and set t is in S, then r has a one-to-one correspondence with t.
If v is a set in S, then the string from 1 to x in Q has a one-to-one correspondence with v.
S has a one-to-one correspondence with the string from 1 to y in Q.
The size of the positive integers in Q is greater than the size of the union of all the sets in S.
The size of the negative integers in Q is greater than the size of the union of all the sets in S.
If x and y are of the same sign, the string from 1 to z in Q has a one-to-one correspondence with the union of all the sets in S.
If x and y are not of the same sign, the string from z to -1 in Q has a one-to-one correspondence with the union of all the sets in S.
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Notation - The product of a pair of integers maybe written this way: x ⋅ y

Definition - Integer z is the product of all of the integers in K if and only if the following statements are true:
E is a path set.
E = (A, B, [C :: B, D], D)
A = Q is an integer set, and z in Q equals 0 in Q, and K is an unordered set; and K is either empty, or K is a subset of Q, or K is coincident with Q.
B = If x is in K, then z becomes the product of x and the previous value of z, and x is removed from K.
C = If K is not empty, repeat B; otherwise continue to D.
D = Stop.
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Definition - Positive integer z equals y raised to the x if and only if the following statements are true:
Q is an integer set containing positive integers x, y and z.
E is a path set.
E = (A, B, [C :: B, D], D)
A = Q is an integer set, and z in Q equals 0 in Q, and K is a non-empty unordered set containing x elements, and the intersection of K and Q is empty.
B = If q is in K, then z becomes the product of y and the previous value of z, and q is removed from K.
C = If K is not empty, repeat B; otherwise continue to D.
D = Stop.
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Notation - Positive integer z equals y raised to the x may be written this way: z = y^x.