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# Exercise 65 - Chapter 1

Give an example of a preorder $$(P,\le)$$ and a subset $$A$$ of its elements such that $$A$$ has more than one meet. In fact, give an example in which $$P$$ only has two elements.

Let $$a,b \in P$$ and subset $$A = \{ a \leq b \; and \; b \leq a \}$$. Then you get two meets namely, $$a \wedge b = a$$ and $$b \wedge a = b$$?
Using reflexivity and transitivity, we can expand this out into a transitivity triangle and get 1. $$a \leq b \leq a$$ and $$a \leq a$$ and 2. $$b \leq a \leq b$$ and $$b \leq b$$. Taking $$a \wedge b$$ in Triangle 1 and Triangle 2 gives a and b respectively.
Comment Source:Let \$$a,b \in P\$$ and subset \$$A = \\{ a \leq b \; and \; b \leq a \\}\$$. Then you get two meets namely, \$$a \wedge b = a\$$ and \$$b \wedge a = b\$$? Using reflexivity and transitivity, we can expand this out into a transitivity triangle and get 1. \$$a \leq b \leq a\$$ and \$$a \leq a\$$ and 2. \$$b \leq a \leq b\$$ and \$$b \leq b\$$. Taking \$$a \wedge b\$$ in Triangle 1 and Triangle 2 gives a and b respectively.