classical logics (the truth table)

We shall call a basic proposition or formula an atom.
For example, an atom is "human beings are animals",
or "sin(x+y)=sin(x)cos(y)+cos(x)sin(y)" and so on.

We can give a true or false value to an atom.
These above are both true.

However, "$y=x+2\quad (x\in [0,1],y\in [0,1])$" is false.

Next we basically introduce 4 connectors and 1 operator for atoms.

(1)A→B : if A,then B.
(2)A・B : A and B.
(3)AＶB : A or B.
(4)A～B : A→B and B→A.
(5)$\neg$A : not A.

By using connectors and operator,
we are able to make a more complicated proposition than an atom.
"A" is "$x$ is a real number."
"B" is "$x$ is a rational number or a irrational number."
We get a new proposition "C" which is "A→B" .
(we have already known "A～B".)

Then, how we get the true or false value of "C" ?

The truth table is gotten.

A B  →
t  t   t
t  f   f
f  t   t
f  f   t

A B  ・
t  t    t
t  f    f
f  t    f
f  f    f

A B   Ｖ　
t  t     t  
t  f     t  
f  t     t  
f  f     f  

A B   ～　
t  t     t  
t  f     f  
f  t     f
f  f     t  

A  $\neg$A　
t    f
f    t