Show that the following conditional statement is a tautology
without
using truth
tables.[
p
^
(
p
!
q
)]
!
q
8. Without the use of truth tables, determine whether (
:
p
^
(
p
!
q
))
!:
q
is a tautology.
show all your work
To determine if the given conditional statement is a tautology without using truth tables, we can use logical equivalences and properties of logical connectives.
Let's break down the problem step by step:
Step 1: Simplify the expression.
We can start by simplifying the inner part of the expression, i.e., `p ^ (p ! q)`.
Using the Distributive Law, we can rewrite it as `p ^ (~q v p)`.
Applying the Commutative Law, we can rearrange the expression as `p ^ (p v ~q)`.
Again, using the Distributive Law, we can simplify it further as `(p ^ p) v (p ^ ~q)`.
The expression `p ^ p` is equivalent to `p`, so we can simplify it to `p v (p ^ ~q)`.
Step 2: Simplify the entire expression.
The given expression is `[(p ^ (p ! q))] ! q`. We will substitute `[p v (p ^ ~q)]` in place of `(p ^ (p ! q))`.
The expression becomes `[p v (p ^ ~q)] ! q`.
Now, let's simplify the expression further using logical equivalences:
[p v (p ^ ~q)] ! q
By the definition of the implication operator (→), we can rewrite the expression as:
! [p v (p ^ ~q)] v q
Using De Morgan's Law, we can apply negation inside the brackets:
(!p ^ ~(p ^ ~q)) v q
By De Morgan's Law again, we can distribute the negation:
(!p ^ (~p v q)) v q
Using the Distributive Law, we distribute the negation further:
[(!p ^ ~p) v (!p ^ q)] v q
Simplifying the expression, we get:
[(F) v (!p ^ q)] v q
Using the identity law, we can simplify `!p ^ q` as `q ^ !p`:
[(F) v q ^ !p] v q
Simplifying further:
(q ^ !p) v q
By the Commutative Law:
q v (q ^ !p)
Using the identity law:
q v q
Simplifying:
q
Step 3: Analyze the final result.
We have simplified the expression to `q`, which shows that the original conditional statement `[(p ^ (p ! q))] ! q` is not a tautology. It is dependent on the value of `q` for its truth value.
Therefore, without using truth tables, we have shown that the given conditional statement is not a tautology.