Multiple Choice

1. Which characteristic MUST be true of a closed mathematical statement?

   It must use only whole numbers.

   It must have exactly one truth value assigned to it: True (T) or False (F).

   It must be an equation involving numbers.

   It must contain at least one variable.

2. Which of the following is considered a statement in logic?

   $10$ is an even number.

   $x + 5 = 12$.

   What is the highest mountain?

   Stop the music!

3. Determine the truth value of the closed statement: "$3 \times 5 = 18$".

   True (T)

   Cannot be determined, as it depends on $3$.

   The sentence is open.

   False (F)

4. Which of the following is an open sentence?

   $5 + 2 < 10$.

   The moon is made of cheese.

   $2 \times 3 = 6$.

   $y$ is a prime number greater than $10$.

5. Given the open sentence $P(x)$: "$x^2 = 25$". If the replacement value for $x$ is $4$, what is the truth value of $P(4)$?

   True (T)

   The sentence remains open.

   False (F)

   Undetermined.

6. The truth value of an open sentence cannot be determined until:

   The variable is replaced by a specific value from the designated replacement set.

   It is evaluated using a calculator.

   It is converted into a question.

   It uses only integers.

7. A closed statement must be assigned exactly how many truth values?

   One (True or False)

   Zero

   Two (True and False simultaneously)

   At least two

8. Determine the truth value of the statement: "If $x=10$, then $x + 3 < 12$."

   True (T)

   The statement is open.

   Undefined logic.

   False (F)

9. What is the primary effect of the negation operator ($\neg$) on a statement $P$?

   It converts $P$ into an open sentence.

   It reverses the meaning and the truth value of $P$.

   It keeps the truth value the same.

   It combines $P$ with another statement $Q$.

10. Let $P$ be the statement: "The square root of $9$ is $3$." ($P$ is True). What is the truth value of $\neg P$?

    Depends on the replacement set.

    True (T)

    Undefined

    False (F)

11. If $S$ is the statement "$5$ is less than $7$", which symbolic expression represents the statement "$5$ is NOT less than $7$"?

    $5 > 7$

    $S \land 7$

    $5 = 7$

    $\neg S$

12. According to the truth table for negation, if a statement $Q$ is False (F), what is the truth value of $\neg Q$?

    Undetermined

    True (T)

    False (F)

    Both True and False

13. Logically, what is the concept of negating a negation, written as $\neg(\neg P)$?

    It must always be False.

    It results in a contradiction.

    It is equivalent to $\neg P$.

    It is logically equivalent to the original statement $P$.

14. If statement $R$ is True (T), what is the resulting truth value of the compound operation $\neg(\neg R)$?

    Undefined

    Depends on the complexity of $R$

    False (F)

    True (T)

15. What is the negation of the inequality statement "$x > 10$"?

    $x < 10$

    $x = 10$

    $x \le 10$

    $x \ne 10$

16. Which symbol represents the logical connective "AND" (Conjunction)?

    $\rightarrow$

    $\neg$

    $\land$

    $\lor$

17. When is the compound statement $P \land Q$ (P AND Q) considered True?

    When both $P$ and $Q$ are False.

    When $P$ is True or $Q$ is True (or both).

    When $P$ is True and $Q$ is False.

    When both component statements $P$ and $Q$ are True.

18. Which symbol represents the logical connective "OR" (inclusive Disjunction)?

    $\land$

    $\neg$

    $\oplus$

    $\lor$

19. When is the compound statement $P \lor Q$ (P OR Q) considered False?

    When $P$ is False and $Q$ is True.

    When $P$ is True and $Q$ is True.

    When $P$ is True and $Q$ is False.

    When both component statements $P$ and $Q$ are False.

20. Let $P$: "$4$ is a prime number" (F). Let $Q$: "$5$ is an odd number" (T). Determine the truth value of $P \land Q$.

    True (T)

    False (F)

    Undetermined

    Both True and False

21. Let $R$: "$2$ is greater than $5$" (F). Let $S$: "$5$ is less than $10$" (T). Determine the truth value of $R \lor S$.

    Cannot be determined

    True (T)

    False (F)

    Only $R$ matters

22. Let $A$: "The number is divisible by $5$." Let $B$: "The number is less than $20$." Write the symbolic form for: "The number is divisible by $5$ OR the number is less than $20$."

    $\neg A$

    $A \rightarrow B$

    $A \land B$

    $A \lor B$

23. Let $C$: "I will travel to Ottawa." Let $D$: "I will take the bus." Write the symbolic form for: "I will travel to Ottawa AND I will take the bus."

    $\neg C$

    $C \lor D$

    $C \land D$

    $C \rightarrow D$

24. If $P$ is True (T) and $Q$ is False (F), what is the truth value of the compound statement $\neg P \lor Q$?

    Undetermined

    False (F)

    True, because $Q$ is False.

    True (T)

25. If $M$ is False (F) and $N$ is True (T), what is the truth value of the compound statement $\neg M \land N$?

    Undetermined

    False (F)

    $N$ is True, so the statement must be False.

    True (T)

26. Let $P(x)$ be the open sentence "$x^2 - 16 = 0$," and let the designated replacement set $R$ be the set of all prime numbers. What is the truth value of the resulting statement when the variable $x$ is replaced by values from $R$?

    The statement is False only if $x=3$.

    The statement is True only for $x=2$.

    The resulting statement is False because the only solutions ($x=4$ and $x=-4$) are not elements of the replacement set $R$.

    The resulting statement is True because $x=4$ is a solution to the equation.

27. Consider the following mathematical expressions: I. $2x + 1 = 7$, where $x$ belongs to the set of natural numbers. II. The number $20$ is divisible by $3$. III. For every even integer $k$, $k^2$ is divisible by $4$. How many of these are classified as closed statements (having a definite truth value)?

    3

    1

    2 (II is a False statement. III is a True statement. I is an open sentence because $x$ is not quantified or replaced.)

    0

28. Let $P(y)$ be the open sentence "$|y - 3| < 1$." If the replacement set for $y$ is $R = \{1, 2, 4, 5\}$, how many distinct values from $R$ make $P(y)$ a True statement?

    1

    4

    3

    None

29. Let $P$ be the statement: "Every perfect square number is strictly greater than $1$." What is the negation $\neg P$?

    Every perfect square number is less than $1$.

    No perfect square number is greater than $1$.

    Every perfect square number is less than or equal to $1$.

    There exists at least one perfect square number that is less than or equal to $1$.

30. If $S$ is the statement "The absolute value of $-5$ is $5$," which of the following expressions is logically equivalent to $S$?

    It is false that the absolute value of $-5$ is $5$.

    It is true that the absolute value of $-5$ is less than $5$.

    The absolute value of $-5$ is greater than $5$.

    It is not the case that it is false that the absolute value of $-5$ is $5$.

31. Statement $Q$: The integer $k$ is a factor of $12$ and $k > 10$. Given that the truth value of $Q$ is False (F), what must be the truth value of $\neg Q$?

    True (T) (The negation operator formally reverses the truth value; $\neg F \equiv T$.)

    True (T) only if $k$ is not a factor of $12$.

    Undetermined, depends on the value of $k$.

    False (F)

32. Let $S$ be the statement: "The sum of the digits of a number $N$ is divisible by $9$." If $N=586$, $S$ is False (F). Which statement correctly describes the truth value of $\neg (\neg (\neg S))$?

    True (T), because $\neg S$ is True.

    Undetermined, depends on whether $5+8+6=19$ is divisible by 9.

    True (T), because $\neg (\neg (\neg S)) \equiv \neg S$. Since $S$ is False, $\neg S$ must be True.

    False (F), because $S$ is False.

33. Let $P$ be "$15$ is a multiple of $5$," and $Q$ be "$15$ is an odd number." What is the truth value of the compound statement $P \land Q$?

    True (T), because $P$ is True, so the entire statement is True.

    False (F), because $P$ is a statement about divisibility and $Q$ is a statement about parity.

    True (T), because $P$ (T) and $Q$ (T) are both True, and $T \land T = T$.

    False (F), because $15$ is a single number.

34. Let $A$ be the statement "$2^3 = 6$," and $B$ be the statement "$3^2 = 9$." What is the truth value of the statement $A \lor B$?

    False (F), because $A$ and $B$ must have the same truth value for disjunction to be True.

    True (T), because $B$ is True, satisfying the condition for inclusive disjunction ($F \lor T = T$).

    Undetermined, as $A$ is False but $B$ is True.

    False (F), because $A$ is False.

35. Let $P$ be "$4 < 6$" and $Q$ be "$6$ is an odd number." Determine the truth value of the statement: "It is not the case that $P$ is False, OR $Q$ is True."

    True (T), because $P$ is True.

    True (T). (Symbolically, this is $\neg (\neg P) \lor Q$. $P$ is T, $\neg P$ is F, $\neg(\neg P)$ is T. $Q$ is F. $T \lor F$ is T.)

    False (F)

    Undetermined, since $Q$ is False.

36. Let $R$ be "$5$ is a prime number" (T) and $S$ be "$10$ is divisible by $4$" (F). Evaluate the truth value of the statement: $\neg (R \land \neg S)$.

    True (T), because $R$ is True.

    True (T)

    False (F). ($S$ is F, so $\neg S$ is T. $R \land \neg S$ is $T \land T = T$. The negation $\neg T$ is False.)

    Undetermined

37. Given that the compound statement $(P \land Q)$ has a truth value of False (F), which of the following scenarios is logically impossible?

    $P$ is False and $Q$ is False.

    $P$ is False and $Q$ is True.

    $P$ is True and $Q$ is True.

    $P$ is True and $Q$ is False.

38. Suppose the statement $(A \lor B)$ is known to be True (T). What definitive truth value must the statement $\neg B$ have?

    $\neg B$ must be False.

    $\neg B$ must be True.

    $\neg B$ could be either True or False, depending on the truth value of $A$. (If $A$ is True, $B$ could be True or False, meaning $\neg B$ could be False or True.)

    $\neg B$ must be the same as the truth value of $A$.

39. Let $P(x)$ be the open sentence "$x$ is an even number" and $Q$ be the closed statement "$5^2 = 10$." If $x$ is replaced by $7$, determine the truth value of the compound statement $P(7) \lor \neg Q$.

    True (T). ($P(7)$ is F. $Q$ is F, so $\neg Q$ is T. $F \lor T$ is True.)

    False (F)

    True (T), because $P(7)$ is True.

    Undetermined, because $P(x)$ was an open sentence.

40. Let $M$: $4 > 3$ (T), $N$: $2+2=5$ (F), and $K$: $1$ is a prime number (F). Evaluate the truth value of the compound statement: $(M \land \neg N) \lor \neg K$.

    True (T). ($\neg N$ is T, $\neg K$ is T. $(T \land T) \lor T \equiv T \lor T \equiv T$.)

    True (T), because $M$ is True.

    False (F), because $N$ and $K$ are both False.

    False (F)