For logic, if there is more than 3 letters ....?

Question

what if a problem had 3 different letters. how would you construct the truth table?

lets use a,b, and c

a would be true, true, false false
then b would be true, false , true , false

but what would c be?

Answer 1

Each additional variable doubles the number of rows: for each of the previous rows you have one copy of the same row with the new variable being true, then another one with the new variable being false.

So you get
a b c
-------
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F

Answer 2

A truth table is just a combination of the variables. For instance , if you have two variables 'a' and 'b' , the truth table will have 4 different combinations - a and b can both be false , a can be false and b can be true , a can be true and b can be false , a and b can both be true.

If you have three variables , the number of combinations is now 8. Just to make it simple consider that with c being false , a and b can go through the above 4 combinations. Also , c being true , a and b can again go through the same 4 combinations.

Basically , the truth table will follow the binary system. Thus if we have 2 variables , we can have values of 0 and 0 , 0 and 1 , 1 and 0 , and 1 and 1.

If there are three variables , the combinations are 000 , 001 , 010 , 011 , 100 , 101 , 110 and 111.

This can be extended to any number of variables. For 'n' variables , the number of possibilities will be 2 raised to the power n.

Answer 3

The truth table has 2^n rows, where n indicates the number of statements (atomic statements, such as p, q, r, s, ...).

It get huge for even small number of statements.

Look, for 5 statements you have 32 rows :-(

Good luck,

Kempos

Answer 4

Kempos is right. truth table has 2^n rows.
Here in your case there are 3 statements a,b,c. so 2^3 = 8 rows.
-----------------------------
a b c
-----------------------------
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
-----------------------------

Hope this will help you.