Q. 2 - Calculus

Question

Logical existence statements: Determine whether each of the statements that follow is true or false. Justify your answers. 

If x is an integer, then there exists some positive integer y such that |y| = x. 

If x is a positive integer, then there exists some negative integer y such that |y| = x.  

If x ∈ [−2, 2], then there exists some y ∈ (0, 4) such that y = x 2. 

If x ∈ [0, 100], then there exists some y ∈ [−10, 10] such that x = y2. 

Step-by-Step Solution

Answer

False.

True.

True.

False.

Step 1. Given information

We have to determine that the given statement is true or false.

Step 2. Explanation

Statement : If x is an integer, then there exists some positive integer y such that |y| = x. 

Suppose x is a negative integer and y is a positive integer then |y| will not be equal to x.

Hence the statement is false.

Statement: If x is a positive integer, then there exists some negative integer y such that |y| = x.  

The given statement is true.

Statement: If x ∈ [−2, 2], then there exists some y ∈ (0, 4) such that y = $x^2$

This statement is true.

If x ∈ [−2, 2] it is very obvious that $x^2\in \left[0,4\right]$. This implies that if x ∈ [−2, 2], then there exist some $y\in (0,4)$ such that $y=x^2$

Statement: If x ∈ [0, 100], then there exists some y ∈ [−10, 10] such that x = $y^2$

This statement is false.

If x ∈ [0, 100] then there exist all $y\in [-10,10]\text{ such that }x-y^2$. There would not be some $y\in [-10,10]$ such that $x=y^2$