To convert the statement "It is false that the pizza is delicious or the cheese is cold" into symbolic form using the given propositions:
- [tex]\( p \)[/tex]: The pizza is delicious.
- [tex]\( q \)[/tex]: The cheese is cold.
Let's break down the statement step-by-step:
1. The disjunction (logical "or") of the two propositions can be written as [tex]\( p \vee q \)[/tex], which means "The pizza is delicious or the cheese is cold."
2. The statement asserts that it is false that this disjunction is true. In logical terms, this can be represented by negating the disjunction [tex]\( p \vee q \)[/tex]. The negation of a disjunction is expressed as [tex]\( \sim(p \vee q) \)[/tex].
Now, let's look at the options and see which one matches [tex]\( \sim(p \vee q) \)[/tex]:
- A. [tex]\( -(p \wedge -q) \)[/tex]: This option involves conjunction (logical "and") and negation in a way that does not match the original statement.
- B. [tex]\( \sim(p \times q) \)[/tex]: This option uses the symbol '×' which is not commonly used for logical disjunction and is not appropriate here.
- C. [tex]\( \sim(p \wedge q) \)[/tex]: This option is the negation of a conjunction, which does not correspond to the negation of a disjunction.
- D. [tex]\( -p \vee q \)[/tex]: This option suggests a disjunction between the negation of [tex]\( p \)[/tex] and [tex]\( q \)[/tex], which does not align with the given statement either.
The correct answer that matches the symbolic form of "It is false that the pizza is delicious or the cheese is cold" is:
B. [tex]\( \sim(p \vee q) \)[/tex]
