Question

23 mark for review
in a certain country, a person must be at least 16 years old to drive a car and must be at least 18 years old
to vote. the variable age represents the age of a person as an integer.
which of the following expressions evaluates to true if the person is old enough to drive but not old
enough to vote, and evaluates to false otherwise?
i. (age ≥ 16) and (age ≤ 18)
ii. (age ≥ 16) and (not(age ≥ 18))
iii. (age < 18) and (not(age < 16))
a i only
b i and ii only
c i and iii only
d ii and iii only

Explanation:

Step1: Analyze the condition

The person must be old enough to drive (age ≥ 16) and not old enough to vote (age < 18, since voting requires age ≥ 18).

Step2: Analyze Expression I

Expression I: \((\text{age} \geq 16) \text{ AND } (\text{age} \leq 18)\)
This means age is between 16 (inclusive) and 18 (inclusive). Since age is an integer, this includes 16, 17, 18. But we need not old enough to vote, i.e., age < 18. However, when age = 18, it is old enough to vote, but in this expression, when age = 18, \((18\geq16)\text{ AND }(18\leq18)\) is true, but 18 is old enough to vote. Wait, but the problem says "old enough to drive but not old enough to vote". So age must be ≥16 and <18. Let's check for age = 16: (16≥16) AND (16≤18) → true. Age = 17: (17≥16) AND (17≤18) → true. Age = 18: (18≥16) AND (18≤18) → true, but 18 can vote, so this expression includes age = 18 which is not desired? Wait, no, the problem says "not old enough to vote" means age < 18. But let's re - examine. Wait, the voting age is at least 18, so not old enough to vote is age < 18. So for expression I: \(\text{age} \geq 16\) and \(\text{age} \leq 18\). Since age is an integer, the values are 16, 17, 18. But 18 is old enough to vote, so when age = 18, the expression I is true, but the person is old enough to vote, which is not what we want. Wait, maybe I made a mistake. Wait, the problem says "old enough to drive but not old enough to vote". So drive: age ≥16, vote: age ≥18, so not vote: age < 18. So the correct range is 16 ≤ age < 18. Now, expression I: \(\text{age} \geq 16\) and \(\text{age} \leq 18\). For age = 18, it is in this range, but 18 can vote. But maybe the problem considers that when age = 18, the person is just old enough to vote, but the expression I is supposed to be true when the person is old enough to drive (≥16) and not old enough to vote (<18). Wait, maybe there is a misinterpretation. Let's check expression II.

Step3: Analyze Expression II

Expression II: \((\text{age} \geq 16) \text{ AND } (\text{NOT}(\text{age} \geq 18))\)
\(\text{NOT}(\text{age} \geq 18)\) is equivalent to \(\text{age} < 18\). So the expression becomes \((\text{age} \geq 16) \text{ AND } (\text{age} < 18)\), which is exactly the condition we need (old enough to drive (≥16) and not old enough to vote (<18)).

Step4: Analyze Expression III

Expression III: \((\text{age} < 18) \text{ AND } (\text{NOT}(\text{age} < 16))\)
\(\text{NOT}(\text{age} < 16)\) is equivalent to \(\text{age} \geq 16\). So the expression becomes \((\text{age} < 18) \text{ AND } (\text{age} \geq 16)\), which is the same as \((\text{age} \geq 16) \text{ AND } (\text{age} < 18)\), which is the correct condition.

Wait, earlier analysis of Expression I: Let's take age = 18. \((18\geq16)\text{ AND }(18\leq18)\) is true, but 18 is old enough to vote. But the problem says "not old enough to vote". So why is Expression I considered? Wait, maybe the problem has a typo, or maybe my analysis is wrong. Wait, the problem says "the variable age represents the age of a person as an integer". So when age = 18, the person can vote, so Expression I includes age = 18 which is not desired. But wait, let's re - read the problem: "evaluates to true if the person is old enough to drive but not old enough to vote". Old enough to drive: age ≥16, not old enough to vote: age < 18. So the correct range is 16 ≤ age < 18.

Expression I: \(\text{age} \geq 16\) and \(\text{age} \leq 18\). So age can be 16,17,18. But 18 is old enough to vote, so when age = 18, the expression I is true, but the person is old enough to vote, so it should be false. But wait, maybe the problem considers "not old enough to vote" as age ≤17? No, the voting age is at least 18, so not old enough is age < 18. So age = 18 is old enough to vote. So Expression I is incorrect? But that contradicts. Wait, maybe I made a mistake. Let's check the expressions again.

Wait, Expression I: (age ≥16) AND (age ≤18). So the age values are 16,17,18. For age = 16: drive (yes), vote (no) → true. Age = 17: drive (yes), vote (no) → true. Age = 18: drive (yes), vote (yes) → but the expression I is true, which is wrong. So Expression I is incorrect? But that can't be. Wait, maybe the problem has a different interpretation. Wait, the problem says "not old enough to vote" – maybe "not old enough" means age ≤17, since voting is at least 18. So age < 18. So Expression I: age between 16 and 18 inclusive. But age = 18 is old enough to vote, so Expression I is wrong. But let's check Expression II: (age ≥16) AND (NOT(age ≥18)). NOT(age ≥18) is age < 18. So age ≥16 and age < 18 → 16,17. Correct. Expression III: (age < 18) AND (NOT(age < 16)). NOT(age < 16) is age ≥16. So age < 18 and age ≥16 → 16,17. Correct. Wait, but earlier I thought Expression I includes age = 18, but maybe the problem considers that when age = 18, the person is just old enough to vote, but the question is "old enough to drive but not old enough to vote". So age = 18 is old enough to vote, so Expression I is wrong. But that would mean only II and III are correct. But let's check the options. The options are:

A) I only

B) I and II only

C) I and III only

D) II and III only

Wait, maybe my analysis of Expression I is wrong. Let's re - evaluate. The problem says "old enough to drive but not old enough to vote". Old enough to drive: age ≥16. Not old enough to vote: age < 18 (since voting requires age ≥18). So the correct condition is 16 ≤ age < 18.

Expression I: (age ≥16) AND (age ≤18). So age can be 16,17,18. For age = 18: drive (yes), vote (yes) → the expression I is true, but the person is old enough to vote, so it should be false. So Expression I is incorrect.

Expression II: (age ≥16) AND (NOT(age ≥18)). NOT(age ≥18) is age < 18. So age ≥16 and age < 18 → 16,17. Correct.

Expression III: (age < 18) AND (NOT(age < 16)). NOT(age < 16) is age ≥16. So age < 18 and age ≥16 → 16,17. Correct.

Wait, but that would mean II and III are correct, so option D. But let's check again.

Wait, maybe the problem has a typo, or maybe I misread Expression I. Wait, Expression I: (age ≥16) AND (age ≤18). If age is 18, the person can vote, so the expression I is true, but the person is old enough to vote, so it should be false. So Expression I is incorrect. Expression II and III are correct. So the answer is D.

Wait, but let's check with age = 18 in Expression I: (18 ≥16) AND (18 ≤18) → true, but the person can vote, so the expression I is true when the person is old enough to vote, which is not desired. So Expression I is wrong. Expression II: (18 ≥16) AND (NOT(18 ≥18)) → (true) AND (false) → false. Correct, because 18 can vote. Expression III: (18 < 18) AND (NOT(18 < 16)) → (false) AND (true) → false. Correct. For age = 17: Expression I: (17 ≥16) AND (17 ≤18) → true. Expression II: (17 ≥16) AND (NOT(17 ≥18)) → true AND true → true. Expression III: (17 < 18) AND (NOT(17 < 16)) → true AND true → true. For age = 16: same as 17. For age = 15: Expression I: (15 ≥16) → false, so overall false. Expression II: (15 ≥16) → false, overall false. Expression III: (15 < 18) AND (NOT(15 < 16)) → true AND false → false. So Expression I is true when age = 18, which is wrong, but Expression II and III are true only when age is 16 or 17, which is correct. So the correct expressions are II and III. So the answer is D.

Answer:

D. II and III only