Question

Decide whether each of the following sentences is valid, satisfiable or unsatisfiable: a) Smoke → Fire b) Smoke → पSmoke c) Smoke v Fire v Fire d) (Smoke → Fire) → (Smoke → [Fire)

Answer 1

The logical propositions are analyzed; 'Smoke → Fire' is satisfiable, 'Smoke → पSmoke' is unsatisfiable due to contradiction, 'Smoke v Fire' is satisfiable, and '(Smoke → Fire) → (Smoke → Fire)' is valid as a tautology.

Step-by-step explanation:

The statements given appear to be related to propositions in logical reasoning, which is a topic within mathematics. Let's analyze each one to determine whether it is valid, satisfiable, or unsatisfiable:

a) Smoke → Fire: This is a conditional statement that can be interpreted as "If there is smoke, then there is fire." It does not guarantee smoke just because there is fire, so without additional information, it's considered satisfiable as it can be true if smoke is present alongside fire, but it also allows for no smoke with no fire.

b) Smoke → पSmoke: This statement seems to contain a typo with the symbol 'प'. If we ignore it as instructed, and interpret this as a negation, the statement "Smoke implies not smoke" is a contradiction, so this is unsatisfiable as it can never be true.

c) Smoke v Fire v Fire: Using 'v' to represent logical 'OR', this statement reads "Smoke or Fire or Fire," which is equivalent to "Smoke or Fire." It is satisfiable because the statement is true if either smoke or fire exists.

d) (Smoke → Fire) → (Smoke → [Fire): There seems to be a bracket error in the second half of the sentence. Assuming it's meant to read "Smoke → Fire," then the statement is a tautology (always true) and is thus valid, as it is essentially saying that if 'if smoke implies fire' is true, then 'smoke implies fire' is also true.

Answer 2

The question involves determining the properties of logical propositions. The sentences a) Smoke implies Fire, b) Smoke implies not Smoke (with a suspected typo), c) Smoke or Fire, and d) (If Smoke then Fire) implies (If Smoke then Fire) are analyzed in terms of satisfiability, validity, and unsatisfiability. Logical propositions are central in the analysis of these statements.

Step-by-step explanation:

The question pertains to logical propositions and their properties such as validity, satisfiability, and unsatisfiability. Here's an analysis of each statement:

a) Smoke → Fire: This is a conditional statement, often read as 'If smoke, then fire.' It's satisfiable because if smoke is false, the implication is true, and if smoke is true and fire is true, the implication is also true.

b) Smoke → ਚSmoke: It appears there's a typo in this statement, but if we interpret 'ਚ' as 'not', the statement reads 'If smoke, then not smoke,' which is unsatisfiable because smoke cannot be both true and not true at the same time.

c) Smoke v Fire v Fire: This is a disjunction, meaning 'Smoke or fire or fire.' It is satisfiable because if at least one of the disjuncts (smoke or fire) is true, the whole statement is true.

d) (Smoke → Fire) → (Smoke → [Fire): If we ignore the symbol '[', this would read 'If (if smoke, then fire), then (if smoke, then fire),' which is a tautology and thus always valid, so it's both valid and satisfiable.

These statements all relate to logical propositions and how we can determine their truth values based on different conditions.