Let P(x) be the statement "Student x knows calculus" and let Q(y) be the statement "Class y contains a student who knows calculus." Express each of these as quantifications of P(x) and Q(y): a) Some students know calculus. b) Not every student knows calculus. ¬∀xP(x) c) Every class has a student in it who knows calculus. ∀yQ(y) Every student in every class knows calculus. ∀x∀yP(x)∧Q(y) There is at least one class with no students who know calculus. ∃y¬Q(y).
Added by Joshua C.
Close
Step 1
So, we can write it as ∃x P(x). Show more…
Show all steps
Your feedback will help us improve your experience
Dr Harish Viswanathan and 66 other Algebra educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Let P(x) be the statement "Student x knows calculus" and let Q(y) be the statement "Class y contains a student who knows calculus." Express each of the following statements as quantifications of P(x) and Q(y): (a) Some students know calculus. (b) Not every student knows calculus. (c) Every class has a student in it who knows calculus. (d) Every student in every class knows calculus. (e) There is at least one class with no students who know calculus.
Vincenzo Z.
Let $P(x, y)$ be the statement "Student $x$ has taken class $y,$ where the domain for $x$ consists of all students in your class and for $y$ consists of all computer science courses at your school. Express each of these quantifications in English. $$ \begin{array}{ll}{\text { a) } \exists x \exists y P(x, y)} & {\text { b) } \exists x \forall y P(x, y)} \\ {\text { c) } \forall x \exists y P(x, y)} & {\text { d) } \exists y \forall x P(x, y)} \\ {\text { e) } \forall y \exists x P(x, y)} & {\text { f) } \forall x \forall y P(x, y)}\end{array} $$
The Foundations: Logic and Proofs
Nested Quantifiers
Let $Q(x, y)$ be the statement "x has sent an e-mail message to $y,$ " where the domain for both $x$ and $y$ consists of all students in your class. Express each of these quantifications in English. $$ \begin{array}{ll}{\text { a) } \exists x \exists y Q(x, y)} & {\text { b) } \exists x \forall y Q(x, y)} \\ {\text { c) } \forall x \exists y Q(x, y)} & {\text { d) } \exists y \forall x Q(x, y)} \\ {\text { e) } \forall y \exists x Q(x, y)} & {\text { f) } \forall x \forall y Q(x, y)}\end{array} $$
Recommended Textbooks
Elementary and Intermediate Algebra
Algebra and Trigonometry
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD