# Monotone function: exercises

Try these ex­er­cises and be­come a de­ity of mono­ton­ic­ity.

## Mono­tone composition

Let $$P, Q$$, and $$R$$ be posets and let $$f : P \to Q$$ and $$g : Q \to R$$ be mono­tone func­tions. Prove that their com­po­si­tion $$g \circ f$$ is a mono­tone func­tion from $$P$$ to $$R$$.

## Evil twin

Let $$P$$ and $$Q$$ be posets. A func­tion $$f : P \to Q$$ is called an­ti­tone if it re­verses or­der: that is, $$f$$ is an­ti­tone when­ever $$p_1 \leq_P p_2$$ im­plies $$f(p_1) \geq_Q f(p_2)$$. Prove that the com­po­si­tion of two an­ti­tone func­tions is mono­tone.

## Par­tial monotonicity

A two ar­gu­ment func­tion $$f : P \times A \to Q$$ is called par­tially mono­tone in the 1st ar­gu­ment when­ever $$P$$ and $$Q$$ are posets and for all $$a \in A$$, $$p_1 \leq_P p_2$$ im­plies $$f(a, p_1) \leq_Q f(a, p_2)$$. Like­wise a 2-ar­gu­ment func­tion $$f : A \times P \to Q$$ is called par­tially mono­tone in the sec­ond ar­gu­ment when­ever $$P$$ and $$Q$$ are posets and for all $$a \in A$$, $$p_1 \leq_P p_2$$ im­plies $$f(p_1, a) \leq_Q f(p_2, a)$$.

Let $$P, Q, R$$, and $$S$$ be posets, and let $$f : P \times Q \to R$$ be a func­tion that is par­tially mono­tone in both of its ar­gu­ments. Fur­ther­more, let $$g_1 : S \to P$$ and $$g_2 : S \to Q$$ be mono­tone func­tions.

Prove that the func­tion $$h : S \to R$$ defined as $$h(s) \doteq f(g_1(s), g_2(s))$$ is mono­tone.

## Brain storm

List all of the com­monly used two ar­gu­ment func­tions you can think of that are par­tially mono­tone in both ar­gu­ments. Also, list all of the com­monly used two ar­gu­ment func­tions you can think of that are par­tially mono­tone in one ar­gu­ment and par­tially an­ti­tone in the other.

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