Question

1. each bus route lists the stops in the order at which they are visited by the bus.
2. each route visits at least four distinct stops.
3. no route visits the same stop twice, except for the first stop, which is always the same as the last stop.
4. there is a stop called downtown, which is visited by each route.
5. every stop other than downtown is visited by at most two routes.

the city has stops at downtown, king st, maxwell ave, stadium district, state st, grace blvd, and charlotte ave. are the following three routes a model for the axiomatic system? justify your argument.
route 1: downtown, king st, stadium district, state st, downtown
route 2: stadium district, state st, grace blvd, maxwell ave, downtown, stadium district
route 3: king st, stadium district, downtown, maxwell ave, stadium district, king st

the three routes
select choice visits select choice a model for the axiomatic system. axioms select choice are satisfied. axiom 3 select choice satisfied because st select choice visits select choice twice and it is not the first/last stop. axiom select choice is select choice because all three routes to the stadium district.

Explanation:

Step1: Check Axiom 1 (Route Lists Stops)

Each route (Route 1: Downtown, King St, Stadium District, State St, Downtown; Route 2: Stadium District, State St, Grace Blvd, Maxwell Ave, Downtown, Stadium District; Route 3: King St, Stadium District, Downtown, Maxwell Ave, Stadium District, King St) lists stops in order. So Axiom 1 is satisfied.

Step2: Check Axiom 2 (At Least Four Distinct Stops)

Route 1: Downtown, King St, Stadium District, State St (4 distinct). Route 2: Stadium District, State St, Grace Blvd, Maxwell Ave, Downtown (5 distinct). Route 3: King St, Stadium District, Downtown, Maxwell Ave (4 distinct). All have ≥4, so Axiom 2 holds.

Step3: Check Axiom 3 (No Stop Twice Except First/Last)

• Route 1: First/Last = Downtown, repeats Downtown (only first/last). Other stops: King St, Stadium District, State St (no repeats).
• Route 2: First/Last = Stadium District, repeats Stadium District (only first/last). Other stops: State St, Grace Blvd, Maxwell Ave, Downtown (no repeats).
• Route 3: First/Last = King St, repeats King St (only first/last). Other stops: Stadium District, Downtown, Maxwell Ave (no repeats). So Axiom 3 is satisfied.

Step4: Check Axiom 4 (Downtown Visited by Each)

Route 1: Downtown (yes), Route 2: Downtown (yes), Route 3: Downtown (yes). Axiom 4 holds.

Step5: Check Axiom 5 (Most Two Routes per Stop)

• Downtown: Route 1, Route 2, Route 3 (3 routes) → Violates Axiom 5 (should be at most 2).

Since Axiom 5 is violated, the three routes are NOT a model for the axiomatic system.

Answer:

No, the three routes are not a model. Axiom 5 is violated (Downtown is visited by 3 routes, exceeding "at most two").