High School Mathematics Extensions/Logic/Solutions

Compound truth tables exercises
1. NAND: x NAND y = NOT (x AND y) 2. NOR: x OR y = NOT (x OR y) 3. XOR: x XOR y is true if and ONLY if either x or y is true.

Produce truth tables for: 1. xyz 2. x'y'z' 3. xyz + xy'z 4. xz 5. (x + y)' 6. x'y' 7. (xy)' 8. x' + y'

Laws of Boolean algebra exercises
1.
 * 1. z = ab'c' + ab'c + abc

\begin{matrix} x &=& ab'c' + ab'c + abc\\ &=& ab'c' + c(ab' + ab)\\ &=& ab'c' + ca \end{matrix} $$
 * 2. z = ab(c + d)

\begin{matrix} x &=& ab(c + d)\\ &=& abc + abd\\ \end{matrix} $$
 * 3. z = (a + b)(c + d + f)

\begin{matrix} x &=& (a + b)(c + d + f)\\ &=& ac + ad + af + bc + bd + bf\\ \end{matrix} $$
 * 4. z = a'c(a'bd)' + a'bc'd' + ab'c

\begin{matrix} x &=& a'c(a'bd)' + a'bc'd' + ab'c\\ &=& a'c(a + (bd)') + a'bc'd' + ab'c\\ &=& a'ca + a'c(bd)' + a'bc'd' + ab'c\\ &=& a'c(b' + d') + a'bc'd' + ab'c\\ &=& a'cb' + a'cd' + a'bc'd' + ab'c\\ \end{matrix} $$
 * 5. z = (a' + b)(a + b + d)d'

\begin{matrix} x &=& (a' + b)(a + b + d)d'\\ &=& (a' + b)(a + b + d)d'\\ &=& (a'a + a'b + a'd + ba + bb + bd)d'\\ &=& (a'b + a'd + ba + b + bd)d'\\ &=& (b(a' + a) + a'd + b + bd)d'\\ &=& (a'd + b + bd)d'\\ &=& a'dd' + bd' + bdd'\\ &=& bd'\\ \end{matrix} $$ 2. Show that x + yz is equivalent to (x + y)(x + z)

\begin{matrix} x &=& (x + y)(x + z)\\ &=& xx + yx + xz + yz\\ &=& x(x + y + z) + yz\\ &=& x + yz\\ \end{matrix} $$

Implications exercises

 * 1) Decide whether the following propositions are true or false:
 * 2) If 1 + 2 = 3, then 2 + 2 = 5 is false because something that's true implies something that's false
 * 3) If 1 + 1 = 3, then fish can't swim is true because 1+1 is not 3
 * 4) Show that the following pair of propositions are equivalent
 * 5) $$x \Rightarrow y$$ : $$y' \Rightarrow x'$$
 * We use truth tables for this
 * The columns in the table are the same for both propositions, thus they are equivalent.
 * The columns in the table are the same for both propositions, thus they are equivalent.

Logic Puzzles exercises
Please go to Logic puzzles.