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    • ▾Precalc
      • ▸Conic sections
        • •Intro to conic sections
        • •Center and radii of an ellipse
        • •Foci of an ellipse
        • •Intro to hyperbolas
        • •Foci of a hyperbola
        • •Maximizing the product of two integers, given their integer sum
        • •Hyperbolas not centered at the origin
        • •Standard equation of a circle
      • ▸Functions and relations
        • •Composition of functions
        • •Operations with functions
        • •Graphing the sum of trig functions from their graphs
        • •Finding and verifying the inverse of a nonlinear function
        • •Graphically finding the increasing, decreasing, and constant intervals
        • •Vertical and horizontal line tests
        • •Restrict the domain then find the inverse
        • •Sketching the inverse of a function from its graph
      • ▸Characteristics of functions
        • •Finding extrema graphically
        • •End behavior
      • ▸Rational functions
        • •Partial fraction decomposition
        • •Simplifying the difference quotient
        • •Pick's theorem
      • ▸Sigma notation and the binomial theorem
        • •Converting sums between sigma notation and expanded form
        • •Properties of sigma notation
        • •Using the binomial theorem
      • ▾Logic
        • •De Morgan's laws for propositional logic
     › Precalc › Logic

    De Morgan's laws for propositional logic

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

    Watch these videos by William Spaniel:

    • DeMorgan's Law, Part 1
    • DeMorgan's Law, Part 2

    Conclude by giving your students these challenges:

    • Calendar Capers by NRICH
    • Colour Building by NRICH
    • Substitution Transposed by NRICH

    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:

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