P → q with NOR operator and Constant F (false)

In summary, we can show that ##p → q## is equivalent to ##F \downarrow ((F \downarrow p) \downarrow q)## by using the properties of the connectives ##\downarrow## and ##∨##. This can also be done with ##T## instead of ##F##.
  • #1
SamitC
36
0
How can I show (without using truth table) that p q is equivalent to F ↓ ((F ↓ p) ↓ q) where F is constant "false" and p and q are propositions?
Is it possible to have a similar kind of expression with T (true) instead of F?
Thanks in advance!
 
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  • #2
##p → q ⇔ \lnot \, p ∨ q## and ##a \downarrow b ⇔ \lnot \, (a \, ∨\, b) = \lnot \, a \, ∧ \lnot \, b.## Thus
$$\begin{align*}
F \downarrow [ ( F \downarrow p) \downarrow q \, ] =&\, F \downarrow [ (\lnot \, F \, ∧ \lnot \, p) \downarrow q \,]\\
=&\, F \downarrow [ \lnot (\lnot \, F \, ∧ \lnot \, p) ∧ \lnot q \,]\\
=&\, F \downarrow [ (F ∨ p) ∧ \lnot q \,]\\
=&\, \lnot F ∧ \lnot [ (F ∨ p) ∧ \lnot q \,]\\
=&\, T ∧ [\lnot (F ∨ p) ∨ q \,]\\
=&\, (\lnot F ∧ \lnot p ) ∨ q \,\\
=&\, (T ∨ q) ∧ (\lnot p ∨ q) \,\\
=&\, T ∧ (\lnot p ∨ q)\,\\
=&\, \lnot p ∨ q\\
=&\, (\, p → q\, )
\end{align*}
$$

As ##T = \lnot \, F## you may simply substitute them.
 

1. What is the truth table for "P → q with NOR operator and Constant F (false)"?

The truth table for "P → q with NOR operator and Constant F (false)" is as follows:

P q P → q NOR Constant F
T T T F F
T F F F F
F T T F F
F F T T F

2. What is the meaning of "P → q with NOR operator and Constant F (false)"?

"P → q with NOR operator and Constant F (false)" means that the statement "P → q" is evaluated using the NOR operator with a constant F (false) value. This means that the statement is true if and only if both P and q are false.

3. How is "P → q with NOR operator and Constant F (false)" different from other logical operators?

"P → q with NOR operator and Constant F (false)" is different from other logical operators because it uses the NOR operator, which is equivalent to the logical operation "NOT OR". This means that the statement is true only when both P and q are false, while other logical operators may have different truth conditions.

4. What is the significance of using the NOR operator and Constant F (false) in "P → q with NOR operator and Constant F (false)"?

The significance of using the NOR operator and Constant F (false) in "P → q with NOR operator and Constant F (false)" is that it allows for the evaluation of the statement "P → q" in a specific context where both P and q are false. This can be useful in certain logical arguments or mathematical proofs.

5. Can "P → q with NOR operator and Constant F (false)" be simplified further?

No, "P → q with NOR operator and Constant F (false)" cannot be simplified further. The NOR operator and Constant F (false) are already the simplest forms in which this statement can be evaluated, as they are the most basic logical operations and values.

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