Adjacency Concept for Pairs of Natural Numbers

The articles [8], [11], [10], [5], [1], [7], [12], [4], [3], [2], [9], and [6] provide the notation and terminology for this paper. In this paper i, j, k, k1, k2, n, m, i1, i2, j1, j2 are natural numbers. Let us consider i1, i2. We say that i1 and i2 are adjacent if and only if: (Def. 1) i2 = i1 + 1 or i1 = i2 + 1. One can prove the following propositions: (1) For all i1, i2 such that i1 and i2 are adjacent holds i1 +1 and i2 +1 are adjacent. (2) For all i1, i2 such that i1 and i2 are adjacent and 1 ≤ i1 and 1 ≤ i2 holds i1 − ′ 1 and i2 − ′ 1 are adjacent. Let us consider i1, j1, i2, j2. We say that i1, j1, i2, and j2 are adjacent if and only if: (Def. 2) i1 and i2 are adjacent and j1 = j2 or i1 = i2 and j1 and j2 are adjacent. The following propositions are true: (3) For all i1, i2, j1, j2 such that i1, j1, i2, and j2 are adjacent holds i1 +1, j1 + 1, i2 + 1, and j2 + 1 are adjacent. (4) Given i1, i2, j1, j2. Suppose i1, j1, i2, and j2 are adjacent and 1 ≤ i1 and 1 ≤ i2 and 1 ≤ j1 and 1 ≤ j2. Then i1 − ′ 1, j1 − ′ 1, i2 − ′ 1, and j2 − ′ 1 are adjacent.

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