Express each of these statements using quantifiers, logical connectives, and Q(x) and P(x). Then express their negations in English and using quantifiers. (a) All students in CMPS... Express each of these statements using quantifiers, logical connectives, and Q(x) and P(x). Then express their negations in English and using quantifiers. (a) All students in CMPS 211 class are CMPS major.
Understand the Problem
The question is asking to express specific statements about students in the CMPS 211 class using logical quantifiers and connectives. It also requires the negation of those statements to be presented in both English and quantified form.
Answer
The statement is: $$ \forall x (P(x)) $$ and its negation is: $$ \exists x (\neg P(x)) $$
Answer for screen readers
The original statement is expressed as: $$ \forall x (P(x)) $$
The negation in quantified form is: $$ \exists x (\neg P(x)) $$
Steps to Solve
- Expressing the statement using quantifiers The statement "All students in the CMPS 211 class are CMPS majors" can be expressed using the universal quantifier. Since the domain is all students in the CMPS 211 class, we use:
$$ \forall x (P(x)) $$
This translates to: "For all students ( x ), ( x ) is a CMPS major."
- Formulating the negation in English The negation of the statement "All students in CMPS 211 class are CMPS majors" is:
"At least one student in the CMPS 211 class is not a CMPS major."
- Expressing the negation using quantifiers The negation can be expressed using the existential quantifier. It is:
$$ \exists x (\neg P(x)) $$
This means: "There exists at least one student ( x ) such that ( x ) is not a CMPS major."
The original statement is expressed as: $$ \forall x (P(x)) $$
The negation in quantified form is: $$ \exists x (\neg P(x)) $$
More Information
This exercise demonstrates how to use logical quantifiers to represent statements about sets of objects, especially in the context of formal logic related to mathematics and computer science. The universal quantifier $\forall$ reflects truths about all members of a set, while the existential quantifier $\exists$ indicates the existence of at least one member that satisfies a condition.
Tips
- Confusing the universal quantifier ($\forall$) with the existential quantifier ($\exists$). Remember that $\forall$ is used for all members, while $\exists$ is used for at least one member.
- Negating the statement incorrectly. The negation of "all" is "not all" or "at least one not," which translates to using the existential quantifier.
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