What is a truth table and what can it do
Mapping a function on a truth table and determining all possible outputs
Applying principles of Boolean Algebra to minimize given function
Learn how to apply minterms and maxterms expansion to a truth table A truth table is basically a representation of all the possible input combinations and the functional output of each of those combinations. It will tell you on a case-by-case basis, what will the functional output be in every input instance.
We need a way to represent the 3 basic logic operations in algebraic formulas. (AND, OR, NOT)
A
⋅
B
=
A
B
{\displaystyle A\cdot B=AB\!}
A
+
B
{\displaystyle A+B\!}
A
¯
{\displaystyle {\overline {A}}\!}
For a simple example, a truth table of the AND gate is given below.
a and b
a
b
f = a b
0
0
0
0
1
0
1
0
0
1
1
1
In general the number of rows in a truth table will be
2
n
,
{\displaystyle 2^{n},\!}
a and b and c
a
b
c
f = a b c
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
We introduce rules of Boolean Algebra to help us deal with simplifying more complex functions. The rules that should be learned are as follows.
Boolean Algebra Theorems and Axioms
Single Variable
Multi-Variable
0
⋅
0
=
0
{\displaystyle 0\cdot 0=0\!}
x
⋅
y
=
y
⋅
x
{\displaystyle x\cdot y=y\cdot x\!}
1
+
1
=
1
{\displaystyle 1+1=1\!}
x
+
y
=
y
+
x
{\displaystyle x+y=y+x\!}
1
⋅
1
=
1
{\displaystyle 1\cdot 1=1\!}
x
⋅
(
y
⋅
z
)
=
(
x
⋅
y
)
⋅
z
{\displaystyle x\cdot (y\cdot z)=(x\cdot y)\cdot z\!}
0
+
0
=
0
{\displaystyle 0+0=0\!}
x
+
(
y
+
z
)
=
(
x
+
y
)
+
z
{\displaystyle x+(y+z)=(x+y)+z\!}
0
⋅
1
=
1
⋅
0
=
0
{\displaystyle 0\cdot 1=1\cdot 0=0\!}
x
⋅
(
y
+
z
)
=
x
⋅
y
+
x
⋅
z
{\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z\!}
1
+
0
=
0
+
1
=
1
{\displaystyle 1+0=0+1=1\!}
x
+
(
y
⋅
z
)
=
(
x
+
y
)
⋅
(
x
+
z
)
{\displaystyle x+(y\cdot z)=(x+y)\cdot (x+z)\!}
If
x
=
0
{\displaystyle x=0\!}
x
¯
=
1
{\displaystyle {\overline {x}}=1\!}
x
+
x
⋅
y
=
x
{\displaystyle x+x\cdot y=x\!}
If
x
=
1
{\displaystyle x=1\!}
x
¯
=
0
{\displaystyle {\overline {x}}=0\!}
x
⋅
(
x
+
y
)
=
x
{\displaystyle x\cdot (x+y)=x\!}
x
⋅
0
=
0
{\displaystyle x\cdot 0=0\!}
x
⋅
y
+
x
⋅
y
¯
=
x
{\displaystyle x\cdot y+x\cdot {\overline {y}}=x\!}
x
+
1
=
1
{\displaystyle x+1=1\!}
(
x
+
y
)
⋅
(
x
+
y
¯
)
=
x
{\displaystyle (x+y)\cdot (x+{\overline {y}})=x\!}
x
⋅
1
=
x
{\displaystyle x\cdot 1=x\!}
x
⋅
y
¯
=
x
¯
+
y
¯
{\displaystyle {\overline {x\cdot y}}={\overline {x}}+{\overline {y}}\!}
x
+
0
=
x
{\displaystyle x+0=x\!}
x
+
y
¯
=
x
¯
⋅
y
¯
{\displaystyle {\overline {x+y}}={\overline {x}}\cdot {\overline {y}}\!}
x
⋅
x
=
x
{\displaystyle x\cdot x=x\!}
x
+
x
¯
⋅
y
=
x
+
y
{\displaystyle x+{\overline {x}}\cdot y=x+y\!}
x
+
x
=
x
{\displaystyle x+x=x\!}
x
⋅
(
x
¯
+
y
)
=
x
⋅
y
{\displaystyle x\cdot ({\overline {x}}+y)=x\cdot y\!}
x
⋅
x
¯
=
0
{\displaystyle x\cdot {\overline {x}}=0\!}
x
⋅
y
+
y
⋅
z
+
x
¯
⋅
z
=
x
⋅
y
+
x
¯
⋅
z
{\displaystyle x\cdot y+y\cdot z+{\overline {x}}\cdot z=x\cdot y+{\overline {x}}\cdot z\!}
x
+
x
¯
=
1
{\displaystyle x+{\overline {x}}=1\!}
(
x
+
y
)
⋅
(
y
+
z
)
⋅
(
x
¯
+
z
)
=
(
x
+
y
)
⋅
(
x
¯
+
z
)
{\displaystyle (x+y)\cdot (y+z)\cdot ({\overline {x}}+z)=(x+y)\cdot ({\overline {x}}+z)\!}
x
¯
¯
=
x
{\displaystyle {\overline {\overline {x}}}=x\!}
Formulate a truth table for the given function below:
f
(
a
,
b
,
c
)
=
a
¯
b
c
+
b
c
+
a
b
¯
c
{\displaystyle f(a,b,c)={\overline {a}}bc+bc+{\overline {ab}}c\!}
