Compound truth tables exercises

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

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

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

Produce truth tables for: 1. xyz

2. x'y'z'

3. xyz + xy'z

4. xz

5. (x + y)'

6. x'y'

7. (xy)'

8. x' + y'

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.

Logic Puzzles exercises

Please go to Logic puzzles.