High School Mathematics Extensions/Logic/Solutions

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Logic[edit]

At the moment, the main focus is on authoring the main content of each chapter. Therefore this exercise solutions section may be out of date and appear disorganised.

If you have a question please leave a comment in the "discussion section" or contact the author or any of the major contributors.


Compound truth tables exercises[edit]

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[edit]

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[edit]

  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[edit]

Please go to Logic puzzles.