Monotonicity and Antimonotonicity

13 Jan 2023

The easiest way to explain monotonicity is to make an analogy with inequalities in algebra. Suppose I had the expression $x \geq y$. Adding any number $a$ to both sides preserves the direction of the inequality. Thus, addition is called monotonic. However, if I multiply by a negative number or take the reciprocals, then the direction of the inequality flips. Thus, these operations are called antimonotonic. We have the same concept in Boolean Algebra with $\Rightarrow$ and $\Leftarrow$.

Monotonic Operations

$(a \Rightarrow b) \quad \Rightarrow \quad (c \wedge a) \Rightarrow (c \wedge b)$

$$\begin{align} &a \Rightarrow b && \\ &= \overline{a} \vee b &&\text{Material Implication} \\ &\Rightarrow (\overline{a} \vee b) \vee \overline{c} &&\text{Generalization} \\ &= \overline{a} \vee (\overline{c} \vee b) &&\text{Associativity and Commutativity} \\ &= \overline{a} \vee (\overline{c} \vee (c \wedge b)) &&\text{Simplification} \\ &= (\overline{c} \vee \overline{a}) \vee (c \wedge b) &&\text{Associativity and Commutativity} \\ &= (\overline{c \wedge a}) \vee (c \wedge b) &&\text{De Morgan's Law} \\ &= (c \wedge a) \Rightarrow (c \wedge b) &&\text{Material Implication} \end{align}$$

$(a \Rightarrow b) \quad \Rightarrow \quad (c \vee a) \Rightarrow (c \vee b)$

$$\begin{align} &a \Rightarrow b && \\ &= \overline{a} \vee b &&\text{Material Implication} \\ &\Rightarrow (\overline{a} \vee b) \vee c &&\text{Generalization} \\ &= (c \vee \overline{a}) \vee b &&\text{Associativity and Commutativity} \\ &= (c \vee (\overline{c} \wedge \overline{a})) \vee b &&\text{Simplification} \\ &= (\overline{c} \wedge \overline{a}) \vee (c \vee b) &&\text{Associativity and Commutativity} \\ &= (\overline{c \vee a}) \vee (c \vee b) &&\text{De Morgan's Law} \\ &= (c \vee a) \Rightarrow (c \vee b) &&\text{Material Implication} \end{align}$$

$(a \Rightarrow b) \quad \Rightarrow \quad (c \Rightarrow a) \Rightarrow (c \Rightarrow b)$

$$\begin{align} &a \Rightarrow b && \\ &= \overline{a} \vee b &&\text{Material Implication} \\ &\Rightarrow (\overline{a} \vee b) \vee \overline{c} &&\text{Generalization} \\ &= (\overline{c} \vee \overline{a}) \vee b &&\text{Associativity and Commutativity} \\ &= (\overline{c} \vee (c \wedge \overline{a})) \vee b &&\text{Simplification} \\ &= (c \wedge \overline{a}) \vee (\overline{c} \vee b) &&\text{Associativity and Commutativity} \\ &= (\overline{\overline{c} \vee a}) \vee (\overline{c} \vee b) &&\text{De Morgan's Law} \\ &= (\overline{c \Rightarrow a}) \vee (c \Rightarrow b) &&\text{Material Implication 2 times} \\ &= (c \Rightarrow a) \Rightarrow (c \Rightarrow b) &&\text{Material Implication} \end{align}$$

Thus, $\wedge$ and $\vee$ are monotonic. $\Rightarrow$ is monotonic in its antecedent.

Antimonotonic Operations

$(a \Rightarrow b) \quad \Rightarrow \quad (a \Rightarrow c) \Leftarrow (b \Rightarrow c)$

$$\begin{align} &a \Rightarrow b && \\ &= \overline{a} \vee b &&\text{Material Implication} \\ &\Rightarrow (\overline{a} \vee b) \vee c &&\text{Generalization} \\ &= (c \vee b) \vee \overline{a} &&\text{Associativity and Commutativity} \\ &= (c \vee (\overline{c} \wedge b)) \vee \overline{a} &&\text{Simplification} \\ &= (b \wedge \overline{c}) \vee (\overline{a} \vee c) &&\text{Associativity and Commutativity} \\ &= (\overline{\overline{b} \vee c}) \vee (\overline{a} \vee c) &&\text{De Morgan's Law} \\ &= (\overline{b \Rightarrow c}) \vee (a \Rightarrow c) &&\text{Material Implication 2 times} \\ &= (c \Rightarrow a) \Rightarrow (c \Rightarrow b) &&\text{Material Implication} \\ &= (a \Rightarrow c) \Leftarrow (b \Rightarrow c) &&\text{Mirror} \end{align}$$

Thus, we see that $\Rightarrow$ is antimonotonic in its consequent.

Note that $=$ and $\neq$ are neither monotonic nor antimonotonic.