High School Mathematics Extensions/Logic/Solutions

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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	Template:Center	Template:Center
0	1	Template:Center	Template:Center
1	0	Template:Center	Template:Center
1	1	Template:Center	Template:Center

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

The NOR function
x	y	x OR y	NOT (x OR y)
0	0	Template:Center	Template:Center
0	1	Template:Center	Template:Center
1	0	Template:Center	Template:Center
1	1	Template:Center	Template:Center

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	Template:Center
0	1	Template:Center
1	0	Template:Center
1	1	Template:Center

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

$\begin{array}{ccc}x& =& a{b}^{\prime }{c}^{\prime }+a{b}^{\prime }c+abc\\ & =& a{b}^{\prime }{c}^{\prime }+c(a{b}^{\prime }+ab)\\ & =& a{b}^{\prime }{c}^{\prime }+ca\end{array}$

2. z = ab(c + d)

$\begin{array}{ccc}x& =& ab(c+d)\\ & =& abc+abd\end{array}$

3. z = (a + b)(c + d + f)

$\begin{array}{ccc}x& =& (a+b)(c+d+f)\\ & =& ac+ad+af+bc+bd+bf\end{array}$

4. z = a'c(a'bd)' + a'bc'd' + ab'c

$\begin{array}{ccc}x& =& a^{\prime }c(a^{\prime }bd{)}^{\prime }+a^{\prime }b{c}^{\prime }{d}^{\prime }+a{b}^{\prime }c\\ & =& a^{\prime }c(a+(bd{)}^{\prime })+a^{\prime }b{c}^{\prime }{d}^{\prime }+a{b}^{\prime }c\\ & =& a^{\prime }ca+a^{\prime }c(bd{)}^{\prime }+a^{\prime }b{c}^{\prime }{d}^{\prime }+a{b}^{\prime }c\\ & =& a^{\prime }c(b^{\prime }+d^{\prime })+a^{\prime }b{c}^{\prime }{d}^{\prime }+a{b}^{\prime }c\\ & =& a^{\prime }c{b}^{\prime }+a^{\prime }c{d}^{\prime }+a^{\prime }b{c}^{\prime }{d}^{\prime }+a{b}^{\prime }c\end{array}$

5. z = (a' + b)(a + b + d)d'

$\begin{array}{ccc}x& =& (a^{\prime }+b)(a+b+d)d^{\prime }\\ & =& (a^{\prime }+b)(a+b+d)d^{\prime }\\ & =& (a^{\prime }a+a^{\prime }b+a^{\prime }d+ba+bb+bd)d^{\prime }\\ & =& (a^{\prime }b+a^{\prime }d+ba+b+bd)d^{\prime }\\ & =& (b(a^{\prime }+a)+a^{\prime }d+b+bd)d^{\prime }\\ & =& (a^{\prime }d+b+bd)d^{\prime }\\ & =& a^{\prime }d{d}^{\prime }+b{d}^{\prime }+bd{d}^{\prime }\\ & =& b{d}^{\prime }\end{array}$

2. Show that x + yz is equivalent to (x + y)(x + z)

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

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. $x\Rightarrow y$ : $y^{\prime }\Rightarrow x^{\prime }$

We use truth tables for this

The NAND function
x	y	$x\to y$	$y^{\prime }\to x^{\prime }$
0	0	Template:Center	Template:Center
0	1	Template:Center	Template:Center
1	0	Template:Center	Template:Center
1	1	Template:Center	Template:Center

The columns in the table are the same for both propositions, thus they are equivalent.

Logic Puzzles exercises

Please go to Logic puzzles.