# High School Mathematics Extensions/Logic/Solutions

## Logic

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### Compound truth tables exercises

1. NAND: x NAND y = NOT (x AND y)

The NAND function
x y x AND y NOT (x AND y)
0 0
0
1
0 1
0
1
1 0
0
1
1 1
1
0

2. NOR: x OR y = NOT (x OR y)

The NOR function
x y x OR y NOT (x OR y)
0 0
0
1
0 1
1
0
1 0
1
0
1 1
1
0

3. XOR: x XOR y is true if and ONLY if either x or y is true.

The XOR function
x y x OR y
0 0
0
0 1
1
1 0
1
1 1
0

Produce truth tables for: 1. xyz

 x y z xyz 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1

2. x'y'z'

 x y z x'y'z' 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 0

3. xyz + xy'z

 x y z xyz + xy'z 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1

4. xz

 x z xz 0 0 0 0 1 0 1 0 0 1 1 1

5. (x + y)'

 x y (x + y)' 0 0 1 0 1 0 1 0 0 1 1 0

6. x'y'

 x y x'y' 0 0 1 0 1 0 1 0 0 1 1 0

7. (xy)'

 x y (xy)' 0 0 1 0 1 1 1 0 1 1 1 0

8. x' + y'

 x y x' + y' 0 0 1 0 1 1 1 0 1 1 1 0

### 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:
1. If 1 + 2 = 3, then 2 + 2 = 5 is false because something that's true implies something that's false
2. If 1 + 1 = 3, then fish can't swim is true because 1+1 is not 3
2. Show that the following pair of propositions are equivalent
1. $x \Rightarrow y$ : $y' \Rightarrow x'$
We use truth tables for this
The NAND function
x y $x \rightarrow y$ $y' \rightarrow x'$
0 0
1
1
0 1
1
1
1 0
0
0
1 1
1
1
The columns in the table are the same for both propositions, thus they are equivalent.