T is an ordered set.

S is either an ordered set or S is a set containing a single element.

Each element in S is also an element in T.

If x and y are in S, and x is associated with y in T, then x is associated with y in S.

If p is associated with q in T, and p is not is S, and q is in S, then there is no element y in S such that y is associated with q.

If q is the starting element of T, and q is in S, then there is no element y in S such that y is associated with q.

If p is associated with q in T, and q is not in S, and p is in S, then p is not associated with any element in S.

If p is the ending element of T, and p is in S, then p is not associated with any element in S.

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T is an ordered set.

S is a string in T.

There is no element in S that is associated with x.

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T is an ordered set.

S is a string in T.

Element z in S is not associated with any element in S.

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Notice that S is either an ordered set or S contains only a single element. S is not empty. If S contains only a single element, then p = q and this element is both the starting element of S and the ending element of S.

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C is an ordered set.

S is an ordered set, or S is a string, or S is a set containing one and only one element.

T is an ordered set, or T is a string, or T is a set containing one and only one element.

The intersection of S and T is empty.

C is the union of S and T.

If x is associated with y in S then x is associated with y in C, and if j is associated with k in T then j is associated with k in C.

If S and T are ordered sets, then the ending element of S in C is associated with the starting element of T in C.

If S contains only a single element e, and T is an ordered set, then e in C is associated with the starting element of T in C.

If S is an ordered set and T contains only a single element d, then the ending element of S in C is associated with d in C.

If S contains only a single element g, and T contains only a single element, then C is an ordered pair and g is its starting element.

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Notice also that the concatenation of sets X and Y is not the same as the concatenation of Y and X. In the latter case, the starting element of the concatenation would be the starting element of Y or Y itself, whereas in the former case, the starting element of the concatenation would be the starting element of X or X itself.

C is a set.

Each element in C is an unordered pair.

Each element in A that is not in the intersection of A and B, and each element in B that is not in the intersection of A and B, is in one and only one unordered pair in C.

Each unordered pair in C contains one and only one element that is also in A, and each unordered pair in C contains one and only one element that is also in B.

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L is an ordered set, and R is an ordered set.

Each element in L and R is not a set.

The intersection of L and R is empty.

L has a one-to-one correspondence with R.

Element q is not a set.

Element q is not in L or R.

C is the concatenation of L and q.

S is the concatenation of C and R.

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Notice that any integer set is an ordered set.

L is an ordered set, and R is an ordered set.

Each element in L and R is not a set.

The intersection of L and R is empty.

L has a one-to-one correspondence with R.

Element q is not a set.

Element q is not in L or R.

C is the concatenation of L and q.

S is the concatenation of C and R.

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L is an ordered set, and R is an ordered set.

Each element in L and R is not a set.

The intersection of L and R is empty.

L has a one-to-one correspondence with R.

Element q is not a set.

Element q is not in L or R.

C is the concatenation of L and q.

S is the concatenation of C and R.

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S = ( a, b, c, q, v, w, x) = (a → b → c → q → v → w → x)

Other notations are possible for

Zero is the center element of an integer set. An integer set is the result of two concatenations. What is eventually defined as the negative integers Q of integer set S is concatenated with an element later called zero. Zero is thereby associated with the ending element of Q. This concatenation is concatenated with what is later defined as the positive integers of S.

Q is an integer set containing integers m and n.

S is a string in Q such that n is the starting element of string S, and the ending element of S is zero.

W is a string in Q such that zero is the starting element of string W, and m is the ending element of string W.

S has a one-to-one correspondence with W.

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T is a non-empty set.

P is the set of positive integers of integer set Q.

T has a one-to-one correspondence with the string from 1 to x in P.

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P is the set of positive integers of integer set Q.

S is a string in P such that the starting element of S is the starting element of P.

V is a string in P such that the starting element of V is the starting element of P.

| S | = | R |.

| V | = | T |.

Element x is in P.

S does not contain x.

V contains x.

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