A set of numbers has the property that for any number t in the set, t + 2 is in...

1. Translate the problem requirements: Understand that we have a set with a specific rule: if any number \(\mathrm{t}\) is in the set, then \(\mathrm{t + 2}\) must also be in the set. We know -1 is in the set, and we need to determine which of the three given numbers (I. -3, II. 1, III. 5) must also be in the set.
2. Apply the forward rule systematically: Starting from -1, repeatedly add 2 to generate all numbers that must be in the set moving in the positive direction.
3. Test the reverse direction: Determine whether the rule forces any numbers less than -1 to be in the set by checking if the rule works backwards.
4. Evaluate each given option: Check whether -3, 1, and 5 appear in our list of numbers that must be in the set based on the rule and starting point.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in simple terms:

• We have a set of numbers with a special rule
• The rule says: if any number \(\mathrm{t}\) is in the set, then \(\mathrm{t + 2}\) must also be in the set
• We know for certain that -1 is in the set
• We need to figure out which of these three numbers must also be in the set: -3, 1, and 5

Think of this like a chain reaction - once we have one number, the rule forces other numbers to be included too.

Process Skill: TRANSLATE - Converting the abstract rule into concrete understanding

2. Apply the forward rule systematically

Since we know -1 is in the set, let's see what other numbers the rule forces us to include by repeatedly adding 2:

Starting with -1:
• -1 is in the set (given)
• \(\mathrm{-1 + 2 = 1}\), so 1 must be in the set
• \(\mathrm{1 + 2 = 3}\), so 3 must be in the set
• \(\mathrm{3 + 2 = 5}\), so 5 must be in the set
• \(\mathrm{5 + 2 = 7}\), so 7 must be in the set
• And this pattern continues forever...

So moving forward (in the positive direction), we get the sequence: \(\mathrm{-1, 1, 3, 5, 7, 9, ...}\)

3. Test the reverse direction

Now let's think about whether any numbers less than -1 must be in the set.

The rule only tells us that "if \(\mathrm{t}\) is in the set, then \(\mathrm{t + 2}\) is also in the set." This is a one-way rule - it doesn't work backwards.

Just because -1 is in the set doesn't mean that \(\mathrm{-1 - 2 = -3}\) has to be in the set. The rule doesn't force us to include numbers by subtracting 2.

To put it another way: if -3 were in the set, then \(\mathrm{-3 + 2 = -1}\) would have to be in the set (which it is). But the fact that -1 is in the set doesn't require -3 to be there.

Process Skill: INFER - Understanding that the rule works in only one direction

4. Evaluate each given option

Now let's check each of the three numbers:

I. -3: We determined that -3 does NOT have to be in the set. The rule doesn't force it to be included based on -1 being in the set.

II. 1: From our forward calculation, 1 MUST be in the set because \(\mathrm{-1 + 2 = 1}\).

III. 5: From our forward calculation, 5 MUST be in the set because we can reach it: \(\mathrm{-1 \to 1 \to 3 \to 5}\).

Therefore, both II and III must be in the set, but I does not have to be.

Final Answer

The numbers that must be in the set are II (1) and III (5), but not I (-3).

This corresponds to answer choice D. II and III only.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the directional nature of the rule
Students often misinterpret the rule "if \(\mathrm{t}\) is in the set, then \(\mathrm{t + 2}\) is in the set" as working in both directions. They incorrectly assume that if -1 is in the set, then \(\mathrm{-1 - 2 = -3}\) must also be in the set. This is a logical error - the rule only works forward (adding 2), not backward (subtracting 2).

2. Confusing "must be" versus "could be" in the set
Students may not distinguish between numbers that are forced to be in the set versus numbers that could potentially be in the set. The question asks which numbers "must also be in the set," meaning they are required by the given rule and starting point, not just possibilities.

3. Overlooking the systematic application needed
Some students may try to analyze each option independently without first systematically determining what the rule generates from the given starting point of -1. They miss that this is a sequential process where each application of the rule leads to the next required number.

Errors while executing the approach

1. Arithmetic mistakes when repeatedly adding 2
When building the sequence \(\mathrm{-1 \to 1 \to 3 \to 5 \to 7...}\), students may make simple arithmetic errors, especially when calculating \(\mathrm{-1 + 2 = 1}\). Getting this first step wrong would cascade into incorrect conclusions about which numbers must be in the set.

2. Stopping the forward progression too early
Students may correctly start with -1 and find that 1 must be in the set, but fail to continue the process to discover that 3 and then 5 must also be in the set. They might miss that 5 is reachable through the chain of applications.

Errors while selecting the answer

1. Including numbers that don't have to be in the set
After determining that 1 and 5 must be in the set, students might still include -3 in their final answer because they incorrectly reasoned that the rule works backward. This leads them to choose answer E (I, II, and III) instead of the correct D (II and III only).

2. Misreading the Roman numeral combinations
Students may correctly identify that 1 and 5 must be in the set (statements II and III) but then select the wrong answer choice due to careless reading of which Roman numerals correspond to which answer option.