When a logical statement is a tautology, the bitstring created when performing all the necessary operations is all 1s. In the following problems we will have two logical variables

*p*and

*q*, so

*p*= 1100 and

*q*= 1010. The AND operator is ^.

The OR operator is v.

The NOT operator is ~.

The IMPLIES operator should be an arrow pointing right, but since that single symbol is not an option in .html, I will use => instead. Remember that

*p*=>

*q*can be changed to ~

*p*v

*q*.

Determine if each of these logical statements is a tautlogy or not.

1)

*p*v (

*p*=>

*q*)

2)

*p*^ (

*p*=>

*q*)

3) (

*p*v ~

*p*) =>

*q*

4) (

*p*^ ~

*p*) =>

*q*

Answers in the comments.

## 1 comment:

Recall that

p= 1100 andq= 1010.1)

pv (p=>q)p=>qis the same as ~pvq.~

pvq=~1100 v 1010 =

0011 v 1010 =

1011

if we take the answer and OR with

p, we get 1100 v 1011 = 1111,so this statement is a tautology.2)

p^ (p=>q)Again,

p=>qbecomes ~pvq.~

pvq=~1100 v 1010 =

0011 v 1010 =

1011

if we take the answer and AND with

p, we get 1100 ^ 1011 = 1000,so this statement is NOT a tautology.3) (

pv ~p) =>q(

pv ~p) =1100 v 0011 =

1111

1111 => 1010 =

~1111 v 1010 =

0000 v 1010 =

1010

This statement is NOT a tautology.4) (p ^ ~p) =>

q(

p^ ~p) =1100 ^ 0011 =

0000

0000 => 1010 =

~0000 v 1010 =

1111 v 1010 =

1111

This statement is a tautology.Post a Comment