Exploring Integer Properties and Equalities

This article delves into the exploration of various statements involving integer properties and equalities. We will rigorously analyze and prove the truth or falsity of each statement, providing detailed mathematical reasoning along the way.

1. For All Integers x, There Exists an Integer y Such That x

Let's consider the statement: For all integers x, there exists an integer y such that x

To test the validity of this, let's consider some specific cases:

Case 1: x ≥ 0. Case 2: x -1. Case 3: x

Case 1: For x ≥ 0, if we take y x 1, then x

Case 2: For x -1, we need to find an integer y such that -1

Case 3: For x

Since the statement fails for x -1, the statement is false.

2. For All Integers x, if x^2 ≠ x, Then There Exists an Integer y Such That x

Consider the statement: For all integers x, if x^2 ≠ x, then there exists an integer y such that x

Proof:

Note that if x^2 ≠ x, then x ≠ 0 and x ≠ 1. If x 1, we can choose y x 1. For the case where 0

Since the statement holds for all cases, the statement is true.

3. For All Integers x, if x^2 x 1, Then x^2 x

Consider the statement: For all integers x, if x^2 x 1, then x^2 x.

Proof:

We start by rearranging the equation:

x^2 x 1

x^2 - x - 1 0

Completing the square:

(x - 0.5)^2 - 0.25 - 1 0

(x - 0.5)^2 1.25

x - 0.5 ±√1.25

x 0.5 ± √1.25

The solutions to this equation are not integers, as √1.25 is not an integer. Hence, the original equation x^2 x 1 cannot be satisfied by any integer x, and the statement is vacuously true (true by default when the premise is false).

Therefore, the statement is true.

4. For All Integers x, if x^2 x and x ≤ 0, Then x is Even

Consider the statement: For all integers x, if x^2 x and x ≤ 0, then x is even.

Proof:

If x^2 x, then x(x - 1) 0. Therefore, x 0 or x 1. Since x ≤ 0, the only possible value is x 0, which is even.

Hence, the statement is true.

Summary

After rigorous analysis, we have determined the truth of each statement as follows:

Statement 1 is false. Statement 2 is true. Statement 3 is true. Statement 4 is true.