Question

Let S(x) denote "x has a good attitude," where the domain is the set of people in the class. Express the negation of the proposition "There is someone in this class who does not have a good attitude" using quantifiers, and then express the negation in English.

Answer 1

See below

Explanation:

Remember, we have two quantifiers, the existential quantifier ∃, and the universal quantifier ∀. The existential ∃ translates to English as "for some" or "there exists", whereas ∀ means "for all" or "every". We will also use the negation operator ¬.

First, let's write the proposition using quantifiers. "There is someone in this class who does not have a good attitude" translates to "(∃x)(¬S(x))". ∃x means that there exists a person in this class x. ¬S(x) means that x, the person that exists because of the quantifier, does not have a good attitude.

The negation is "¬(∃x)(¬S(x))" or equivalently "(∀x)(S(x))". To negate a proposition using quantifiers, change the quantifier (existential to universal and viceversa) and negate the predicate (in this case we negated ¬S(x)).

In English, "(∀x)(S(x))" means "Every person in this class has a good attitude".