The truth values of the propositions are false, true and false
How to determine the true values?
The Boolean variables are given as:
p = false
r = false
q = true
s = true
The truth values of the propositions are as follows:
1. (s → (p ∧ ¬ r)) ∧ ((p → (r ∨ q)) ∧ s)
As a general rule:
∧ means and
¬ not
∨ or
So, we have:
(true → (false ∧ ¬ false)) ∧ ((false → (false ∨ true)) ∧ true)
¬ false= true.
So, we have:
(true → (false ∧ true)) ∧ ((false → (false ∨ true)) ∧ true)
false ∧ true = false and false ∨ true = true
So, we have:
(true → false) ∧ ((false → true) ∧ true)
false → true = true and true → false = false
So, we have:
(false) ∧ (true ∧ true)
true ∧ true = true
So, we have:
false ∧ true
Lastly
false ∧ true = false
Hence, the truth value of (s → (p ∧ ¬ r)) ∧ ((p → (r ∨ q)) ∧ s) is false
2. ((p ∧ ¬ q) → (q ∧ r)) → (s ∨ ¬q)
This means
((false ∧ ¬ true) → (true ∧ false)) → (true ∨ ¬true)
Using the established rules in (1), we have:
((false ∧ false) → (true ∧ false)) → (true ∨ false)
This gives
((false → false)) → true
This gives
true → true
Lastly, we have:
true
Hence, the truth value of ((false ∧ ¬ true) → (true ∧ false)) → (true ∨ ¬true) is true
3. (p → q) ∧ (q → r)
This gives
(false → true) ∧ (true → false)
Using the established rules in (1), we have:
true ∧ false
This gives
false
Hence, the truth value of (p → q) ∧ (q → r) is false
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