Why inclusive or is used with Either...Or... here?

Question

There are 2 propositions

p: It is below freezing
q: It is snowing.

I want to write the symbolic form of: Either it is below freezing or it is snowing, but it is not snowing if it is freezing.

This is what I came up with: $(p \oplus q) \wedge (p \to \neg q)$, but in the answer key it is $(p \vee q) \wedge (p \to \neg q)$.

Second part it clear to me, but in the first shouldn't it will be exclusive or, because of Either...or...?

Comments

I have checked the truth table of both the cases, they are same. Therefore two statement are equivalent.

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The question is the well answered here and here.

Answer 1

The block after the AND statement is ensuring the condition of ⊕

You can understand it like this: 

P $\vee$ Q   ...(1)

P $\rightarrow$ ¬Q   ...(2)

are joined by $ \wedge$

(1) is true when PQ = 01 or 10 or 11

(2) is true when PQ = 00 or 01 or 10

(1) $ \wedge$ (2) takes only {10,01} which is, as you said correctly, the case of X-OR.

We do not have an XOR operator ⊕ in Prepositional Logic.