Let's break down and solve the logical statements based on the universe of discourse $\{1, 2, 3, 4, 5\}$:

Given Statements:

1. The universe of discourse is $\{1, 2, 3, 4, 5\}$.
2. Determine the truth values of the following:
   • a. $(\exists x). x + 1 < 5$
   • b. $(\forall x). x \leq 5$
   • c. $(\forall x). x + 4 < 10$

Solutions:

a. $(\exists x). x + 1 < 5$

• This statement means: "There exists an $x$ in the set $\{1, 2, 3, 4, 5\}$ such that $x + 1 < 5$."
• Let's check each value of $x$:
  • $x = 1 \rightarrow 1 + 1 = 2 < 5$ (True)
  • Since $x = 1$ satisfies the condition, the statement is true.

b. $(\forall x). x \leq 5$

• This statement means: "For all $x$ in the set $\{1, 2, 3, 4, 5\}$, $x \leq 5$."
• Checking all values:
  • $x = 1, 2, 3, 4, 5$, all of these satisfy $x \leq 5$.
• Therefore, the statement is true.

c. $(\forall x). x + 4 < 10$

• This statement means: "For all $x$ in the set $\{1, 2, 3, 4, 5\}$, $x + 4 < 10$."
• Checking each value of $x$:
  • $x = 1 \rightarrow 1 + 4 = 5 < 10$ (True)
  • $x = 2 \rightarrow 2 + 4 = 6 < 10$ (True)
  • $x = 3 \rightarrow 3 + 4 = 7 < 10$ (True)
  • $x = 4 \rightarrow 4 + 4 = 8 < 10$ (True)
  • $x = 5 \rightarrow 5 + 4 = 9 < 10$ (True)
• Since all values satisfy the condition, the statement is true.

Summary of Truth Values:

• a. True
• b. True
• c. True

Would you like any further details or explanations on any part of this solution?

Here are 5 related questions that expand upon this topic:

1. What is the difference between $\exists x$ and $\forall x$ in logic?
2. How would the truth values change if the universe was $\{1, 2, 3, \ldots, 10\}$?
3. Can a statement with $(\exists x)$ be false if at least one value of $x$ satisfies the condition?
4. How can you negate statements with $(\forall x)$ and $(\exists x)$?
5. What other mathematical operations can be used in logical statements besides addition?

Tip: In logic, when you see $(\exists x)$, you only need to find one value that satisfies the condition to make the statement true.