The domain of a relation R is the set of real numbers. x is related to y under relation R if |x+y|≥2. Select the description that accurately describes relation R.

Question

BlackTom AI · Accepted Answer

1. Relevant concepts:
- Reflexive: for every real x, xRx must hold.
- Anti-reflexive (irreflexive): for every real x, xRx must not hold.
- Here xRy is the condition |x + y| ≥ 2. In particular xRx ⇔ |2x| ≥ 2 ⇔ 2|x| ≥ 2 ⇔ |x| ≥ 1.

2. Step-by-step reasoning (reflexivity and anti-reflexivity):
- Check reflexive: reflexivity requires |2x| ≥ 2 for all real x, i.e., |x| ≥ 1 for every x. This is false because some real numbers have |x| < 1. Concrete counterexample: x = 0 gives |0 + 0| = |0| = 0, and 0 ≥ 2 is false. Therefore R is not reflexive.
- Check anti-reflexive: anti-reflexive would require |2x| < 2 for every real x, i.e., |x| < 1 for every x. This is false because some real x have |x| ≥ 1. Concrete counterexample: x = 1 gives |1 + 1| = |2| = 2, and 2 ≥ 2 is true, so xRx holds for x = 1. Thus R is not anti-reflexive.

3. Calculations shown:
- For xRx: |x + x| = |2x| = 2|x|.
- At x = 0: 2|0| = 0 < 2 → not reflexive.
- At x = 1: 2|1| = 2 ≥ 2 → not anti-reflexive.

4. Why the correct answer is correct:
Because there exist some elements (e.g., x = 1) for which xRx is true, and others (e.g., x = 0) for which xRx is false, the relation fails both the "for all" requirement of reflexivity and the "for all not" requirement of anti-reflexivity. Hence it is neither.

5. Why the other options are incorrect:
- Reflexive: incorrect because x = 0 violates xRx.
- Anti-reflexive: incorrect because x = 1 satisfies xRx.
- Transitive: also false (not required for the chosen answer), e.g., x = 0, y = 2, z = 0: |0+2| = 2 (0Ry true), |2+0| = 2 (2R0 true), but |0+0| = 0 < 2 (0R0 false), so transitivity fails.

6. Conclusion:
The relation R is neither reflexive nor anti-reflexive, which matches the provided correct answer.

The domain of a relation R is the set of real numbers. x is related to y under relation R if |x+y|≥2. Select the description that accurately describes relation R.单项选择题

类似问题

Which of the following Math relation is transitive but neither reflexive nor symmetric?

A binary relation R on a single set A is a subset of?

Which of the following Math relation is transitive but neither reflexive nor symmetric?

The domain of a relation R is the set of integers. x is related to y under relation R if x2=y. Select the description that accurately describes relation R.

The domain of a relation R is the set of real numbers. x is related to y under relation R if |x+y|≥2. Select the description that accurately describes relation R.

A relation R is given in the matrix below. Rows of the matrix are numbered 1 through 4 from top to bottom and columns are numbered 1 through 4 from left to right. Select the expression that is false.

Let A={a,b}, B={c,d,e}. The total number of possible relations from A to B is?

Select the set that corresponds to the relation given in the matrix below. Rows of the matrix are numbered 1 through 4 from top to bottom and columns are numbered 1 through 4 from left to right.

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单