Discrete Mathematics - Arguments with Quantified Statements

Question

I'd like to know whether my argument proof is sufficient or whether my rule of inference is correct for the following.

1. A photographer in ABC tour group has not visited Opera House
2. Every photographer in ABC tour group has visited Botanic Garden

Q: Can you conclude that "Someone in ABC tour group has not visited Opera House"?

Argument Proof:

1. Let $T$ be the set of all photographers in ABC tour group.
2. Let $O(t)$ be “$t$ visited Opera House”.
3. Let $B(t)$ be “$t$ visited Botanic Garden”.
4. $\exists t \in T, (\neg O(t)).$ (Premise)
5. $\forall t \in T, (B(t)).$ (Premise)
6. $\exists t \in T, (B(t) → ~O(t)).$ (By Universal Modus Ponens) Q.E.D.

Any help is greatly appreciated. Thank you.

Answer 1

Let X be the set of people in ABC tour group.

Using the first sentence,
$\exists$x, P(x) $\wedge$ $\sim$ O(x)
Then using existential instantiation,
P(a) $\wedge$ $\sim$ O(a)
Then using specialisation,
$\sim$ O(a)
Finally using existentiation generalisation, you will get the answer.

It seems you do not need the second sentence. Do ask your tutor for clarification.