# 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
${\displaystyle {\begin{matrix}x&=&ab'c'+ab'c+abc\\&=&ab'c'+c(ab'+ab)\\&=&ab'c'+ca\end{matrix}}}$
2. z = ab(c + d)
${\displaystyle {\begin{matrix}x&=&ab(c+d)\\&=&abc+abd\\\end{matrix}}}$
3. z = (a + b)(c + d + f)
${\displaystyle {\begin{matrix}x&=&(a+b)(c+d+f)\\&=&ac+ad+af+bc+bd+bf\\\end{matrix}}}$
4. z = a'c(a'bd)' + a'bc'd' + ab'c
${\displaystyle {\begin{matrix}x&=&a'c(a'bd)'+a'bc'd'+ab'c\\&=&a'c(a+(bd)')+a'bc'd'+ab'c\\&=&a'ca+a'c(bd)'+a'bc'd'+ab'c\\&=&a'c(b'+d')+a'bc'd'+ab'c\\&=&a'cb'+a'cd'+a'bc'd'+ab'c\\\end{matrix}}}$
5. z = (a' + b)(a + b + d)d'
${\displaystyle {\begin{matrix}x&=&(a'+b)(a+b+d)d'\\&=&(a'+b)(a+b+d)d'\\&=&(a'a+a'b+a'd+ba+bb+bd)d'\\&=&(a'b+a'd+ba+b+bd)d'\\&=&(b(a'+a)+a'd+b+bd)d'\\&=&(a'd+b+bd)d'\\&=&a'dd'+bd'+bdd'\\&=&bd'\\\end{matrix}}}$

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

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

### 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. ${\displaystyle x\Rightarrow y}$  : ${\displaystyle y'\Rightarrow x'}$
We use truth tables for this
The NAND function
x y ${\displaystyle x\rightarrow y}$  ${\displaystyle y'\rightarrow x'}$
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.