High School Mathematics Extensions/Logic/Solutions

LogicEdit

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Compound truth tables exercisesEdit

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 exercisesEdit

1.

1. z = ab'c' + ab'c + abc
 
2. z = ab(c + d)
 
3. z = (a + b)(c + d + f)
 
4. z = a'c(a'bd)' + a'bc'd' + ab'c
 
5. z = (a' + b)(a + b + d)d'
 

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

 

Implications exercisesEdit

  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.   :  
We use truth tables for this
The NAND function
x y    
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 exercisesEdit

Please go to Logic puzzles.