
Before we experiment with circuits we need to decide how to
record what happens when we change the switches in the circuit from
0 to 1 or from 1 to 0. We use small letters $p$ and $q$ to
represent the switches. These switches also represent statements
that are true or false. A true statement has truth value 1 and a
false statement has truth value 0.
In circuit diagrams each switch is either 'on' (representing
the number 1 or a true statement) or 'off' (representing the number
0 or a false statement). Combinations of switches called
logical gates represent
the logical connectives
.

Logic is concerned with compound statements formed from given
statements by means of the
connectives 'not', 'and', 'or', 'if
... then' and combinations of these connectives. For example "Today
is Tuesday" is a simple statement and "The sun is shining" is a
simple statement. Both statements can be true or false. Combining
these two statements will make a compound statement, for example
"Today is Tuesday and the sun is shining".
Logical arguments are a combination of statements, and circuits are
a combination of switches that control the flow of current. Modern
computers store data in the form of '0's and '1's, and perform many
operations on that data, including arithmetic. Studying the
relation between truth tables and circuits will help us to
understand a little of the underlying principles behind the design
of computers.
Fig. 1 gives a summary of the information you need to build
your own circuits. 
Fig. 1

First, connect a switch to a lamp as in Fig. 2 and click on the
switch several times to change it from 1 to 0 and back to 1.
Observe that the light goes on when the switch registers 1 and the
light goes off when the switch registers 0. 
Fig. 2

Fig. 3

Now put a NOT gate into the circuit between the switch and the
lamp, as shown in Fig. 3, and observe what happens when you change
the switch from 1 to 0. 
Given a statement $p$, the
negation not$p$, written $\neg p$, is
demonstrated by this circuit and defined by the following truth
table. $$ \begin{array}[ p & \neg p \\ 1 & 0 \\ 0 & 1
\end{array} $$
And
The logical connective 'and' is used to make a compound statement
from two simpler statements. For example if we speak the truth when
we say "Today is Friday and it is raining" then "Today is Friday"
must be true and " it is raining" must be true. If either or both
are false then the compound statement is false.
Make and test the circuit shown in Fig. 4 and fill in the truth
table replacing question marks by 1 if the light goes on and zero
if the light does not go on. 
Fig. 4

$$ \begin{array}[ p &q & p\wedge q \\ 1 &1 &?\\
1 &0 &?\\ 0 &1 &?\\ 0 &0 &? \\ \end{array}
$$ 
If you want to check your work then you can click
here.
Or
In everyday language we often use 'or' inclusively as in "If you
want to buy cereals or soft drinks go to aisle 7 in the
supermarket" when we expect people who want to buy both cereals and
soft drinks to go to that aisle as well as people wanting to buy
just one of them. On other occasions we use 'or' exclusively to
offer two alternatives expecting the listener to choose just one of
them as in "Do you want steak for dinner or chicken?". The meaning
is usually clear from the context but to avoid any ambiguity in
mathematics the logical connective 'or' is always used inclusively
and never exclusively.
Given any two statements$p$, $q$ then $p$ or $q$, (written $p
\vee q$)is given in the truth table on the right. Make the circuit
shown in Fig. 5 and replace the question marks by 1's and 0's
according to whether the lamp lights up or not. 
Fig. 5 
$$ \begin{array}[ p &q & p\vee q \\ 1 &1 &?\\ 1
&0 &?\\ 0 &1 &?\\ 0 &0 &? \\ \end{array}
$$ 
Again you can check your work by clicking
here.
XOR, NAND, NOR and
XNOR
Now experiment with the circuits for XOR, NAND, NOR and XNOR and
replace the question marks in the following truth table. $$
\begin{array}[ p &q &p XOR q & p NAND q &p NOR q
& p XNOR q \\ 1 &1 &? &? &? &? \\ 1 &0
&? &? &? &? \\ 0 &1 &? &? &? &?
\\ 0 &0 &? &? &? &? \\ \end{array} $$ Comparing
these tables to what we have already discovered we see that p NOR q
means 'not(p or q)'. Also p NAND q means 'not(p and q)'.
You can check the truth tables for XOR, NAND, NOR and XNOR by
clicking
here.
If you want to learn more about logic and circuits then read
Logic, Truth Tables and Swirching Circuits which covers the
content of this article and goes further to discuss tautologies,
implication and the Sheffer stroke.