Final answer:
Either Sidney will fail without studying hard, or Sidney will fail even if Quincy studies hard and passes. Symbolically, this can be represented as:
Sidney fails ⇔ Sidney does not study hard
Sidney fails ∨ (Quincy studies hard ∧ Sidney passes)
Explanation:
In this statement, we have two possible outcomes for Sidney. Either he fails his exams regardless of Quincy's performance, or he fails only if Quincy studies hard and still manages to pass his own exams. This can be represented using logical operators in symbolic form.
Let's start by defining the propositions:
- P: Sidney passes his exams
- Q: Sidney studies hard
- R: Quincy passes his exams
Using these propositions, we can represent the given statement in symbolic form:
P ⇔ Q
P ∨ (Q ∧ R)
The first expression is an equivalence statement. It says that Sidney passing his exams is equivalent to him studying hard. In other words, if Sidney passes his exams, then he must have studied hard, and if he studied hard, then he must have passed his exams. This is represented using the double arrow (⇔) operator.
The second expression is a disjunction statement with a conjunction in its scope. It says that either Sidney fails his exams or both Quincy and Sidney pass their exams. This is represented using the double vertical bar (∨) operator for disjunction and the double dot (∧) operator for conjunction.
To simplify this expression, we can use the distributive law to distribute the disjunction over the conjunction:
P ∨ (Q ∧ R) = P ∨ Q ∨ R - (Q ∧ R) implies P (De Morgan's law)
= P ∨ Q - (Q ∧ R) implies P - Q implies R (Double negation law)
= P - Q implies R - P implies R (De Morgan's law again)
= (P implies R) and (R implies P) (Implication equivalence)
= P <=> R (Biconditional equivalence)
= Sidney passes if and only if Quincy passes.