The question asks to determine the logical values (truth or falsity) of each statement given that the universe of discourse is the set of real numbers. Let's analyze each statement:

a. $(\forall x) \, 2x + 1 = 2x$

• This statement means that for all $x$, the equation $2x + 1 = 2x$ must hold. Simplifying the equation: $2x + 1 = 2x \implies 1 = 0$ This is clearly false for all real numbers. Hence, the statement is false.

b. $(\exists x) \, 2x + 1 = 2x$

• This statement means that there exists some $x$ such that $2x + 1 = 2x$. From the analysis in part (a), we know that the equation reduces to $1 = 0$, which is impossible for any real number. Therefore, this statement is also false.

c. $(\forall x) \, |x| = x$

• This statement asserts that for all $x$, $|x| = x$. We know that: $|x| = x \quad \text{only if} \quad x \geq 0$ However, for negative $x$, $|x| = -x$, so the statement does not hold for all real numbers. Therefore, this statement is false.

d. $(\exists x) \, |x| = x$

• This means there exists at least one $x$ such that $|x| = x$. This is true for any $x \geq 0$. For example, $x = 1$ satisfies this condition. Therefore, this statement is true.

e. $(\exists x) \, |x| = 0$

• This means there exists an $x$ such that $|x| = 0$. We know that $|x| = 0$ only if $x = 0$. Therefore, this statement is true since $x = 0$ satisfies the condition.

f. $(\exists x) \, x^2 = x$

• This means there exists an $x$ such that $x^2 = x$. Simplifying the equation: $x^2 = x \implies x(x - 1) = 0$ So, $x = 0$ or $x = 1$. Since there are solutions, the statement is true.

Summary of Results:

a. False
b. False
c. False
d. True
e. True
f. True

Would you like more details on any of these solutions or have any other questions?

Here are 5 related questions to expand on the concepts:

1. What is the difference between universal and existential quantifiers in logic?
2. How do absolute values affect equations with variables?
3. Can you explain why $|x| = x$ is true only for non-negative values of $x$?
4. How can we solve equations involving $x^2 = x$ and similar quadratic forms?
5. What are the implications of contradictory statements like $2x + 1 = 2x$ in logical reasoning?

Tip: When solving logical statements, always pay close attention to whether the statement is universal (applies to all values) or existential (applies to at least one value).