Final answer:
The logical expression for the sentence provided is "(!p ∨ r) → !q", using the propositional variables given in the question.
Step-by-step explanation:
The sentence "If you do not meet the maximum weight requirement or are younger than 18 years old, then you may not join the military." can be translated into a logical expression using the propositions provided. Since "you do not meet the maximum weight requirement" is the negation of "p", we can denote it as "!p". The proposition "you are younger than 18 years old" is already given as "r".
The phrase "you may not join the military" is the negation of "q", so it can be expressed as "!q". Therefore, the logical expression formed from the given sentence is "(!p ∨ r) → !q", which means if either proposition !p (not meeting the weight requirement) or proposition r (being younger than 18) is true, then proposition !q (not being able to join the military) follows.
To express the sentence "If you do not meet the maximum weight requirement or are younger than 18 years old, then you may not join the military" as a logical expression, we need to use propositional logic.
We can translate the sentence as follows:
[(¬P) ∨ R] → ¬Q
P represents the proposition "you meet the maximum weight requirement"
Q represents the proposition "you may join the military"
R represents the proposition "you are younger than 18 years old"
In the logical expression, ¬ represents the negation (not) operator, ∨ represents the disjunction (or) operator, and → represents the implication (if...then) operator.