The problem you provided is asking to determine the truth values of three logical propositions involving the set of numbers $\{1, 2, 3, 4, 5\}$. Let's analyze each statement.

The universe of discourse is the set $\{1, 2, 3, 4, 5\}$.

1. (a) $(\exists x) \, x + 1 < 5$
   This statement is asking if there exists an $x$ in the set $\{1, 2, 3, 4, 5\}$ such that $x + 1 < 5$.

   We will test each element:

   • For $x = 1$: $1 + 1 = 2$, and $2 < 5$.
   • For $x = 2$: $2 + 1 = 3$, and $3 < 5$.
   • For $x = 3$: $3 + 1 = 4$, and $4 < 5$.
   • For $x = 4$: $4 + 1 = 5$, and $5 \not< 5$.
   • For $x = 5$: $5 + 1 = 6$, and $6 \not< 5$.

   Therefore, at least one $x$ satisfies the condition (in fact, $x = 1, 2, 3$ all work). So the statement is true.

2. (b) $(\forall x) \, x \leq 5$
   This statement asserts that for all $x$ in the set $\{1, 2, 3, 4, 5\}$, $x \leq 5$.

   Testing all elements:

   • $1 \leq 5$
   • $2 \leq 5$
   • $3 \leq 5$
   • $4 \leq 5$
   • $5 \leq 5$

   All elements satisfy the condition, so the statement is true.

3. (c) $(\forall x) \, x + 4 < 10$
   This statement asserts that for all $x$ in the set $\{1, 2, 3, 4, 5\}$, $x + 4 < 10$.

   We will test each element:

   • For $x = 1$: $1 + 4 = 5$, and $5 < 10$.
   • For $x = 2$: $2 + 4 = 6$, and $6 < 10$.
   • For $x = 3$: $3 + 4 = 7$, and $7 < 10$.
   • For $x = 4$: $4 + 4 = 8$, and $8 < 10$.
   • For $x = 5$: $5 + 4 = 9$, and $9 < 10$.

   All elements satisfy the condition, so the statement is true.

Summary:

• (a) is true.
• (b) is true.
• (c) is true.

Would you like more details on any specific part? Here are some related questions for practice:

1. How would the truth value of (a) change if the universe of discourse was $\{3, 4, 5\}$?
2. What would happen to statement (b) if we changed the upper bound to 4?
3. How would statement (c) change if we modified the expression to $x + 4 \leq 9$?
4. How can we express (b) in negation form?
5. Can we generalize the truth of (b) and (c) for larger sets like $\{1, 2, \ldots, 10\}$?

Tip: In logic, testing the truth of quantified statements often involves verifying examples for existential quantifiers and checking all cases for universal quantifiers.