Which of the following statements are true? (x denotes a real number). (A) ?x(x > 2 -> x^2 > 4) (B) ?x(x > 2 ? x^2 > 4) (C) ?x(x > 2 ? x^2 > 4) Select one: a. (B) only b. (A) and (C) c. All of them d. (C) only e. (A) only
Added by Carla H.
Close
Step 1
The implication x > 2 -> x^2 > 4 means that if x is greater than 2, then x^2 must be greater than 4. This is true because any real number greater than 2 when squared will give a value greater than 4. Therefore, statement (A) is true. Show more…
Show all steps
Your feedback will help us improve your experience
Darshan Maheshwari and 88 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
Determine the truth value of each of these statements if the domain consists of all real numbers. $$ \begin{array}{ll}{\text { a) } \quad \exists x\left(x^{3}=-1\right)} & {\text { b) } \exists x\left(x^{4}<x^{2}\right)} \\ {\text { c) } \quad \forall x\left((-x)^{2}=x^{2}\right)} & {\text { d) } \forall x(2 x>x)}\end{array} $$
Linda H.
Vincenzo Z.
Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. $$ \begin{array}{ll}{\text { a) } \forall x \exists y\left(x^{2}=y\right)} & {\text { b) } \forall x \exists y\left(x=y^{2}\right)} \\ {\text { c) } \exists x \forall y(x y=0)} & {\text { d) } \exists x \exists y(x+y \neq y+x)}\end{array} $$ $$ \begin{array}{l}{\text { e) } \forall x(x \neq 0 \rightarrow \exists y(x y=1))} \\ {\text { f) } \exists x \forall y(y \neq 0 \rightarrow x y=1)} \\ {\text { g) } \forall x \exists y(x+y=1)} \\ {\text { h) } \exists x \exists y(x+2 y=2 \wedge 2 x+4 y=5)} \\ {\text { i) } \forall x \exists y(x+y=2 \wedge 2 x-y=1)} \\ {\text { j) } \forall x \forall y \exists z(z=(x+y) / 2)}\end{array} $$
The Foundations: Logic and Proofs
Nested Quantifiers
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