It seems you have uploaded a file. I will extract and analyze its contents to assist you. Please hold on for a moment. PIL import Image

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image_path = '/mnt/data/1733557427960wuysggi4.jpg' img = Image.open(image_path)

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img.show()It seems that the image has been loaded, but I cannot directly interpret its contents. Let me now process and extract any mathematical data or text it may contain. Please hold on for a moment. pytesseract

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extracted_text = pytesseract.image_to_string(img)

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extracted_textThe image contains a question about a relation $R$ defined on the set of real numbers $\mathbb{R}$. The relation $R$ is given by:

$(x, y) \in R \text{ if } |xy| < 1$

The task is to determine whether the relation $R$ is:

1. Transitive but not symmetric
2. Symmetric but not transitive
3. Both transitive and symmetric
4. Neither transitive nor symmetric

Solution

Symmetry:

A relation is symmetric if for every pair $(x, y) \in R$, it also holds that $(y, x) \in R$.

For $R$, we are given that $|xy| < 1$. If $(x, y) \in R$, then $|xy| < 1$, and since multiplication is commutative (i.e., $xy = yx$), it follows that $|yx| = |xy|$. Hence, if $(x, y) \in R$, then $(y, x) \in R$ as well.

Thus, the relation is symmetric.

Transitivity:

A relation is transitive if whenever $(x, y) \in R$ and $(y, z) \in R$, it also holds that $(x, z) \in R$.

Let's analyze the transitivity condition. We have $(x, y) \in R$ and $(y, z) \in R$, which means:

$|xy| < 1 \quad \text{and} \quad |yz| < 1$

We need to check whether this implies $|xz| < 1$.

Consider specific values: let $x = 0.5$, $y = 0.5$, and $z = 0.5$. We have:

$|xy| = |0.5 \times 0.5| = 0.25 < 1 \quad \text{and} \quad |yz| = |0.5 \times 0.5| = 0.25 < 1$

But for $(x, z)$:

$|xz| = |0.5 \times 0.5| = 0.25 < 1$

However, if we take larger values like $x = 1$, $y = 0.5$, and $z = 1$, we get:

$|xy| = |1 \times 0.5| = 0.5 < 1 \quad \text{and} \quad |yz| = |0.5 \times 1| = 0.5 < 1$

But for $|xz|$:

$|xz| = |1 \times 1| = 1 \quad \text{(not less than 1)}$

This counterexample shows that the relation is not transitive.

Conclusion:

The correct answer is:

(b) Symmetric but not transitive.

Would you like further clarification or details on how transitivity is checked? Let me know if you have any questions!

Additional Questions:

1. What does symmetry mean in the context of relations?
2. Can you provide another example of a relation that is symmetric but not transitive?
3. What other properties can a relation have, besides symmetry and transitivity?
4. How can we prove whether a relation is transitive using different examples?
5. What are some real-world scenarios where such relations might arise?

Tip: Always test a few specific cases when checking properties like transitivity, as counterexamples can often provide clear insight into whether the property holds.