To restate and solve your question carefully: we need to find which logical statement represents "It is cold and humid if and only if it is snowing."
Let's denote the following:
- [tex]\(p\)[/tex]: It is cold.
- [tex]\(q\)[/tex]: It is humid.
- [tex]\(r\)[/tex]: It is snowing.
First, we need to understand the structure of "if and only if" (denoted as [tex]\(\leftrightarrow\)[/tex]). The statement "It is cold and humid if and only if it is snowing" can be expressed logically as:
[tex]\[ (p \wedge q) \leftrightarrow r \][/tex]
This statement means:
- If it is cold and humid, then it is snowing.
- If it is snowing, then it is cold and humid.
Now let's match this logical statement to one of the provided expressions:
1. [tex]\((p \vee g) \rightarrow -r\)[/tex]: This means "Either it is cold or g (an undefined variable) implies not r (it is not snowing)", which is not related to our statement.
2. [tex]\((p-1) \wedge(q-1)\)[/tex]: This one has syntax that does not conform to logical operations involving [tex]\(p, q,\)[/tex] and [tex]\(r\)[/tex].
3. [tex]\( \neg \wedge a) - n \wedge |r| b \wedge a b\)[/tex]: This expression is not clearly structured in logical terms involving [tex]\(p, q, r\)[/tex].
4. [tex]\( (p \rightarrow r) \wedge (q \rightarrow r) \wedge (r \rightarrow p) \wedge (r \rightarrow q) \)[/tex]: Although this expression signifies multiple conditions, it doesn't directly map to our double implication.
So with careful consideration, none of the provided options accurately or clearly represent [tex]\((p \wedge q) \leftrightarrow r \)[/tex].
It seems there might be a typographical or syntactical error in the options provided. However, the most appropriate and correct logical expression for "It is cold and humid if and only if it is snowing" should indeed be [tex]\((p \wedge q) \leftrightarrow r\)[/tex].
