Here are four propositions. Which are true and which false? Justify your answers. (a) ∀x ∈ R, ∃y ∈ R such that y 4 = 4x. (b) ∃y ∈ R such that ∀x ∈ R we have y 4 = 4x. (c) ∀y ∈ R, ∃x ∈ R such that y 4 = 4x. (d) ∃x ∈ R such that ∀y ∈ R we have y 4 = 4x

(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.