Let x1 and x2 be 0-1 variables whose values indicate whether projects 1 and 2 are not/are completed. Which of the following answers indicates that project 2 can be completed only if project 1 has been completed?

Question

BlackTom AI · Accepted Answer

x2 = 1 ⇒ x1 = 1, so the linear constraint must force x2 ≤ x1. Rewriting gives x2 − x1 ≤ 0, which is the correct constraint.

1. Relevant knowledge points and concepts
- Binary (0-1) variables: xi = 1 means project i is completed; xi = 0 means not completed.
- Logical implication for binaries: “project 2 can be completed only if project 1 is completed” is the implication x2 = 1 ⇒ x1 = 1.
- Converting implication to a linear inequality for 0-1 variables: x2 ≤ x1 is equivalent to x2 − x1 ≤ 0.

2. Step-by-step reasoning
- Start from the English condition: if x2 = 1 then x1 must equal 1.
- For binary variables the inequality x2 ≤ x1 enforces exactly that: x2 can be 1 only when x1 is 1; if x2 = 0, x1 can be 0 or 1.
- Rearranging x2 ≤ x1 yields x2 − x1 ≤ 0, which matches one of the answer choices.

3. Intermediate steps / small proof by cases (truth table)
List all four possible (x1,x2):
- (0,0): x2 − x1 = 0 − 0 = 0 ≤ 0 → allowed.
- (0,1): x2 − x1 = 1 − 0 = 1 ≤ 0 → violated (disallowed), correctly preventing project 2 without project 1.
- (1,0): x2 − x1 = 0 − 1 = −1 ≤ 0 → allowed.
- (1,1): x2 − x1 = 1 − 1 = 0 ≤ 0 → allowed.
Only (0,1) is disallowed, which matches the intended rule.

4. Why the correct answer is correct
x2 − x1 ≤ 0 is algebraically equivalent to x2 ≤ x1, which enforces the implication x2 = 1 ⇒ x1 = 1 for 0-1 variables. It blocks the only forbidden case (x1 = 0, x2 = 1) while permitting all legally allowed combinations.

5. Why the other options are incorrect
- x1 + x2 = 0: forces both projects not completed (both must be 0), which is too restrictive.
- x1 + x2 = 1: forces exactly one project to be completed (exclusive), which disallows (1,1) even though completing both should be allowed.
- x2 − x1 ≥ 0 (equivalently x2 ≥ x1): allows x2 = 1 with x1 = 0, the opposite of the required implication.

6. Conclusion
The constraint x2 − x1 ≤ 0 correctly encodes “project 2 can be completed only if project 1 has been completed.” This matches the provided answer: x2 − x1 ≤ 0.

Let x1 and x2 be 0-1 variables whose values indicate whether projects 1 and 2 are not/are completed. Which of the following answers indicates that project 2 can be completed only if project 1 has been completed?单项选择题

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