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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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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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.

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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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 has a one-to-one correspondence with T .

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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.

T does not contain x.

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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 does not contain y.

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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

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.

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

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. When non-zero terms are added, the sum changes. Precisely defining such a process requires definitions that use the terms

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 P 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 elements in P.

Element x is associated with one and only one element at any given time.

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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.Within the brackets (within [x ::

Notice that if j is the element in P such that no other element in P is associated with it, then j necessarily does not appear in any list in brackets. Also, if k is not associated with any element in P, it may appear in a list in brackets.

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

A = (q, r, s, t, u, v, w, [x ::

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

B = (q, r, s, [t ::

E is an event 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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E is an event set.

E = (A, B, [C ::

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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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 v is a set in S, then the string from 1 to the absolute value of x in Q has a one-to-one correspondence with v.

S has a one-to-one correspondence with the string from 1 to the absolute value of y in Q.

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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E is an event set.

E = (A, B, [C ::

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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E is an event set.

E = (A, B, [C ::

A = Q is an integer set containing positive integer x and integer y; and z in Q equals 0, and K is a non-empty unordered set containing x elements, and the intersection of K and Q is empty.

B = If any element 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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We now have opportunities to write equations that contain more than one operation. To avoid confusion as to which order the operations are intended to be calculated. there is a conventional use of parentheses that is illustrated below:

z = (x ⋅ y + 1) + y

The calculations indicated within a pair of parentheses is to be done before any other operations are done with the contents of the parentheses. Within the parentheses, or in a string of operations without parentheses, addition and subtraction are done last. The order would be this:

z = (x ⋅ y + 1) + y

= Q + y

= Q + R - 1

= M - 1

= N