If the operation @ is defined for all integers a and b by a@b = a + b - ab,...

1. Translate the problem requirements: We have a custom operation @ defined as \(\mathrm{a@b = a + b - ab}\). We need to check three properties: (I) commutativity (does order matter?), (II) identity with 0 (does \(\mathrm{a@0}\) equal a?), and (III) associativity (does grouping matter?).
2. Test commutativity by comparing \(\mathrm{a@b}\) with \(\mathrm{b@a}\): Substitute the definition into both expressions and see if they're algebraically identical.
3. Test the zero identity property: Substitute \(\mathrm{b = 0}\) into the operation definition and simplify to see if we get back to a.
4. Test associativity by comparing \(\mathrm{(a@b)@c}\) with \(\mathrm{a@(b@c)}\): Calculate each grouped expression step by step and verify they produce the same result.
5. Determine which statements are always true: Based on our verification of each property, identify which Roman numerals must be true for all integers.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're dealing with here. We have a custom operation called @ that works differently from regular addition or multiplication. When we write \(\mathrm{a@b}\), it means we calculate \(\mathrm{a + b - ab}\).

We need to check three properties:

- Property I asks: Does \(\mathrm{a@b}\) always equal \(\mathrm{b@a}\)? (This is like asking if 3+5 equals 5+3, but for our @ operation)
- Property II asks: Does \(\mathrm{a@0}\) always equal a? (This is like asking if 0 acts like an "identity" - leaving a unchanged)
- Property III asks: Does \(\mathrm{(a@b)@c}\) always equal \(\mathrm{a@(b@c)}\)? (This is about whether grouping matters, like how (2+3)+4 equals 2+(3+4) in regular addition)

Process Skill: TRANSLATE - Converting the abstract operation definition into concrete testable properties

2. Test commutativity by comparing \(\mathrm{a@b}\) with \(\mathrm{b@a}\)

Let's see if switching the order of a and b makes any difference.

Using our definition \(\mathrm{a@b = a + b - ab}\):

- \(\mathrm{a@b = a + b - ab}\)
- \(\mathrm{b@a = b + a - ba}\)

Now, since regular addition gives us \(\mathrm{a + b = b + a}\), and regular multiplication gives us \(\mathrm{ab = ba}\), we can see that:
\(\mathrm{b@a = b + a - ba = a + b - ab = a@b}\)

So \(\mathrm{a@b = b@a}\) is always true! Let's verify with a concrete example: if a = 3 and b = 2:

- \(\mathrm{3@2 = 3 + 2 - (3×2) = 5 - 6 = -1}\)
- \(\mathrm{2@3 = 2 + 3 - (2×3) = 5 - 6 = -1}\) ✓

Property I is TRUE.

3. Test the zero identity property

Let's see what happens when we operate any number a with 0.

Using our definition with b = 0:
\(\mathrm{a@0 = a + 0 - a×0 = a + 0 - 0 = a}\)

This always gives us back a! Let's check with an example: if a = 5:
\(\mathrm{5@0 = 5 + 0 - (5×0) = 5 + 0 - 0 = 5}\) ✓

Property II is TRUE.

4. Test associativity by comparing \(\mathrm{(a@b)@c}\) with \(\mathrm{a@(b@c)}\)

This is the trickiest one. We need to carefully work through both sides step by step.

Left side: \(\mathrm{(a@b)@c}\)
First, calculate \(\mathrm{a@b = a + b - ab}\)
Then, calculate \(\mathrm{(a@b)@c = (a + b - ab) + c - (a + b - ab)×c}\)
Expanding: \(\mathrm{= a + b - ab + c - c(a + b - ab)}\)
\(\mathrm{= a + b - ab + c - ca - cb + abc}\)
\(\mathrm{= a + b + c - ab - ca - cb + abc}\)

Right side: \(\mathrm{a@(b@c)}\)
First, calculate \(\mathrm{b@c = b + c - bc}\)
Then, calculate \(\mathrm{a@(b@c) = a + (b + c - bc) - a×(b + c - bc)}\)
Expanding: \(\mathrm{= a + b + c - bc - a(b + c - bc)}\)
\(\mathrm{= a + b + c - bc - ab - ac + abc}\)
\(\mathrm{= a + b + c - ab - ac - bc + abc}\)

Comparing both results:

- Left side: \(\mathrm{a + b + c - ab - ca - cb + abc}\)
- Right side: \(\mathrm{a + b + c - ab - ac - bc + abc}\)

Since \(\mathrm{ac = ca}\) and \(\mathrm{bc = cb}\), these expressions are identical!

Property III is TRUE.

Process Skill: MANIPULATE - Careful algebraic expansion and comparison of complex expressions

5. Determine which statements are always true

We've verified that all three properties hold:

- Property I (commutativity): TRUE
- Property II (zero identity): TRUE
- Property III (associativity): TRUE

Therefore, "I, II and III" must all be true.

Final Answer

The answer is E: "I, II and III". All three properties of the @ operation are true for all integers a, b, and c. The operation is commutative, has 0 as an identity element, and is associative.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what needs to be tested: Students may think they need to find specific values of a, b, and c that make the statements true, rather than testing whether the statements are true for ALL integers. This leads them to plug in numbers without understanding they need to prove the properties algebraically.

2. Confusion about the operation definition: Students may misread \(\mathrm{a@b = a + b - ab}\) as \(\mathrm{a@b = a + b + ab}\) or struggle to consistently apply the subtraction of the product term, leading to incorrect calculations throughout.

3. Not recognizing standard algebraic properties: Students may not immediately recognize that they're being asked to test commutativity, identity, and associativity properties, missing the systematic approach needed to verify each property.

Errors while executing the approach

1. Algebraic expansion errors in associativity: When testing \(\mathrm{(a@b)@c = a@(b@c)}\), students commonly make mistakes in the multi-step expansion, especially when distributing \(\mathrm{(a + b - ab) × c}\) or \(\mathrm{a × (b + c - bc)}\), leading to incorrect terms or signs.

2. Premature conclusion from limited examples: Students may test the properties with only one or two specific number examples (like a=1, b=2) and conclude the properties are true without doing the complete algebraic proof, missing cases where their examples might not represent all integers.

3. Sign errors in complex expressions: Students frequently make sign mistakes when dealing with multiple negative terms, particularly in expressions like \(\mathrm{a + b - ab + c - c(a + b - ab)}\), incorrectly handling the negative multiplication.

Errors while selecting the answer

1. Partial verification leading to incorrect selection: Students may correctly verify one or two properties but make an error on the third, then select an answer choice that only includes the properties they verified correctly (like 'I and II only') instead of recognizing all three are true.

2. Mismatching verified properties to answer choices: Even when students correctly determine which properties are true, they may incorrectly map their findings to the Roman numeral answer choices, selecting the wrong combination due to careless reading of the options.