High School Mathematics Extensions/Logic/Solutions

High School Mathematics Extensions

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Supplementary Chapters — Primes and Modular Arithmetic — Logic

Mathematical Proofs — Set Theory and Infinite Processes — Counting and Generating Functions — Discrete Probability

Matrices — Further Modular Arithmetic — Mathematical Programming — Markov Chains

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.

Logic Puzzles exercises

Please go to Logic puzzles.