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
