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When defining set, we mentioned that repetition and order are irrelevant for a set in the sense that $\{a,b,c\}=\{b,c,a,b,b,c\}$. This somewhat tells us the existence of order for the elements of sets. In fact, for many sets we are interested in, one or more orders are assumed in the common sense so that we can compare those elements. For example, for the set $\{1,2,3,4\}$, there is apparently an order called ‘less than’ (小于) which means we can ‘compare’ two elements in the manner whether one number is less than the other number or not. Similarly ‘less or equal to’ (小于等于), ‘greater than’ (大于), ‘greater or equal to’ (大于等于), and ‘equal to’ (等于) are other four orders for this set.

There is a mathematical way to define an ‘order’ for a set. This founds an important cornerstone for mathematical logic, and it is the main task of this post.

1. Ordered pair (有序对)