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Immediate Successor – Predecessor of Natural Numbers

Hi, infinite-set reader. Are you ready to Set Theory?

This time we will discuss about immediate successor and immediate predecessor and its properties. Keep in mind that x^{+} is x\cup\left\{ x\right\}. Happy reading.


If the immediate successor of two natural numbers is equal then the two natural numbers are the same.


Take n,m\in\omega such that n^{+}=m^{+}. Will be proven that n=m. Suppose n\neq m. Because of n\in n^{+}=m^{+} then n\in m or n=m. On the other hand, because of m\in m^{+}=n^{+} then m\in n or m=n. According to assumptions n\neq m then it is only occur n\in m and m\in n. According to the Theorem (x) in this link, we get n\subseteq m and m\subseteq n equivalent to n=m. There was a contradiction then it should be n=m.

In the previous discussion has introduced the notion of immediate successor. Using the same sense, can be recovered an element n if it has been known to the immediate successor of n. Furthermore n is called the immediate predecessor of the immediate successor of n. In other words, n is called immediate predecessor of natural numbers m if n^{+}=n\cup\left\{ n\right\} =m.