Consider the following:
1. [tex]$(\neg Z \vee \neg I) \leadsto \neg(H \wedge Q)$[/tex]
2. [tex]$(\neg Z)$[/tex]
[tex]$\therefore(L \rightarrow N)$[/tex]
Which of the following can be derived from the premises?
a) [tex]$(L)$[/tex]
b) [tex]$(-2 \vee -4)$[/tex]
c) [tex]$(- H \vee C )$[/tex]
Answer (2)
None of the options (a, b, or c) can be derived from the given premises. We find that ¬ H ∨ ¬ Q can be derived, but none closely match the provided options. Thus, the answer is that no options can be derived from the premises.
;
¬ Z is true from premise 2.
( ¬ Z ∨ ¬ I ) is true because ¬ Z is true.
¬ ( H ∧ Q ) is true because ( ¬ Z ∨ ¬ I ) ⟹ ¬ ( H ∧ Q ) .
None of the options a, b, or c can be derived from the premises. Therefore, the answer is that none of the options can be derived. There seems to be a typo in options b and c. Assuming option c was meant to be ¬ H ∨ ¬ Q , then we could derive that option. But as it is, we cannot derive any of the options.
Explanation
Understanding the Problem We are given two premises:
( ¬ Z ∨ ¬ I ) ⟹ ¬ ( H ∧ Q )
¬ Z
We want to determine which of the given options can be derived from these premises.
The options are: a) L b) − 2 ∨ − 4 c) − H ∨ C )
Analyzing Premise 2 From premise 2, we know that ¬ Z is true. This means that the expression ( ¬ Z ∨ ¬ I ) is also true, regardless of the truth value of ¬ I , because if ¬ Z is true, then the disjunction ( ¬ Z ∨ ¬ I ) is true.
Applying Premise 1 Since ( ¬ Z ∨ ¬ I ) is true, and we know from premise 1 that ( ¬ Z ∨ ¬ I ) ⟹ ¬ ( H ∧ Q ) , we can conclude that ¬ ( H ∧ Q ) is true.
Using De Morgan's Law Using De Morgan's law, we can rewrite ¬ ( H ∧ Q ) as ( ¬ H ∨ ¬ Q ) . This means that either H is false, or Q is false, or both are false.
Evaluating the Options Now let's examine the given options: a) L : We cannot derive anything about L from the given premises. The conclusion ∴ ( L → N ) is not a premise, and we cannot assume it is true. b) − 2 ∨ − 4 : This option is nonsensical in the context of logical statements and cannot be derived from the premises. c) − H ∨ C ) : We derived ¬ H ∨ ¬ Q . This is not the same as ¬ H ∨ C . We cannot derive this from the premises.
Conclusion Therefore, none of the options can be directly derived from the premises.
Examples
In logic and reasoning, deriving conclusions from premises is similar to solving a puzzle where you use given clues to find the solution. For example, if you know that 'If it is raining, then the ground is wet' and you also know that 'It is raining', you can conclude that 'The ground is wet'. This type of logical deduction is used in computer science for program verification, in mathematics for proving theorems, and in everyday life for making informed decisions.
