Students will be introduced to De Morgan's laws, and see a proof of each law using truth tables.

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Conclude by giving your students these challenges:

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Suppose your domain consists of only the natural numbers. Express the functions \(\text{not}(x),\) \(\text{and}(x, y),\) \(\text{or}(x, y),\) in terms of arithmetic: \(+,\) \(-,\) \(\times,\) \(\div.\) 0 denotes false, 1 denotes true. For example, we expect \(\text{not}(0) = 1,\) and \(\text{or}(1, 0) = 1.\) Each function can take in 0 or 1, and spit out 0 or 1. For example, \(\text{not}(3)\) is undefined, as it doesn't have a sensible interpretation.

Here's the answer:

$$\begin{align}
\text{not}(x) &= 1 - x \\
\text{and}(x, y) &= xy \\
\text{or}(x, y) &= \text{not}(\text{and}(\text{not}(x), \text{not}(y))) \\
&= 1 - (1 - x)(1 - y)
\end{align}$$