Answer:
(a) TRUE
(b) FALSE
(c) TRUE
(d) FALSE
Step-by-step explanation:
(a) ∀x ∈ R, ∃y ∈ R such that y^4 = 4x.
For all x ∈ R, there exists y ∈ R such that y^4 = 4x. TRUE.
In this exact order, here is what this statement says:
- Any real number can be chosen as x.
- After x is chosen, we can find at least one y based on x such that y^4 = 4x.
The statement ensures that x is chosen first, and for that x, there is a y that satisfies the equation, which is true.
Example: For x = 4, there exists y = 2, such that 2^4 = 4(4) = 16
(b) ∃y ∈ R such that ∀x ∈ R we have y 4 = 4x.
There exists y ∈ R such that for all x ∈ R, y^4 = 4x. FALSE.
In this exact order, here is what this statement says:
- Atleast one y can be found before any other variable is chosen. And the equation will be satisfied irrespective of what x is.
- After this unique y is chosen, we can choose any x into the equation y^4 = 4x and it will be valid.
This statement is false.
Example, Let y = 2, and x = 3
2^4 ≠ 4(3)
(c) ∀y ∈ R, ∃x ∈ R such that y^4 = 4x.
For all y ∈ R, there exists x ∈ R such that y^4 = 4x.
This statement is in the form of (a) above, but in this case, y is chosen first. The statement is still TRUE.
(d) ∃x ∈ R such that ∀y ∈ R we have y^4 = 4x.
There exists x ∈ R such that for all y ∈ R, y^4 = 4x.
This statement is the form of (b), but in this case, x is chosen first. The statement is FALSE.
