Question

The relation \( R \), defined by \( R = \{ (T_1, T_2) : T_1 \text{ is similar to } T_2 \} \), in the set \( A \) of all triangles, is

• A. reflexive and symmetric, but not transitive
• B. reflexive and transitive, but not symmetric
• C. symmetric and transitive, but not reflexive
• D. reflexive, symmetric and also transitive

Hint:

Relations like "is similar to," "is congruent to," and "is equal to" are classic examples of equivalence relations in mathematics because they satisfy all three conditions: reflexivity, symmetry, and transitivity. Memorizing these examples can save time in exams.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To determine the properties of the relation R, we need to check for reflexivity, symmetry, and transitivity. A relation that possesses all three properties is known as an equivalence relation.
The relation is defined on the set A of all triangles, where (\(T_1\), \(T_2\)) \(\in\) R means that triangle \(T_1\) is similar to triangle \(T_2\).
Step 2: Detailed Explanation:
Reflexivity:
A relation R is reflexive if (a, a) \(\in\) R for every element a in the set.
For any triangle \(T_1\) \(\in\) A, \(T_1\) is always similar to itself (\(T_1\) ~ \(T_1\)). This is because the corresponding angles are equal and the ratio of corresponding sides is 1.
Therefore, (\(T_1\), \(T_1\)) \(\in\) R. The relation is reflexive.
Symmetry:
A relation R is symmetric if (a, b) \(\in\) R implies (b, a) \(\in\) R.
Let (\(T_1\), \(T_2\)) \(\in\) R. This means \(T_1\) is similar to \(T_2\) (\(T_1\) ~ \(T_2\)).
If \(T_1\) is similar to \(T_2\), then it follows that \(T_2\) is also similar to \(T_1\) (\(T_2\) ~ \(T_1\)).
Therefore, (\(T_2\), \(T_1\)) \(\in\) R. The relation is symmetric.
Transitivity:
A relation R is transitive if (a, b) \(\in\) R and (b, c) \(\in\) R implies (a, c) \(\in\) R.
Let (\(T_1\), \(T_2\)) \(\in\) R and (\(T_2\), \(T_3\)) \(\in\) R. This means \(T_1\) is similar to \(T_2\) (\(T_1\) ~ \(T_2\)) and \(T_2\) is similar to \(T_3\) (\(T_2\) ~ \(T_3\)).
If \(T_1\) is similar to \(T_2\), their corresponding angles are equal and the ratio of their corresponding sides is constant.
Similarly, if \(T_2\) is similar to \(T_3\), their corresponding angles are equal and the ratio of their corresponding sides is constant.
It follows that the corresponding angles of \(T_1\) and \(T_3\) are also equal, and the ratio of their corresponding sides is constant. Thus, \(T_1\) is similar to \(T_3\) (\(T_1\) ~ \(T_3\)).
Therefore, (\(T_1\), \(T_3\)) \(\in\) R. The relation is transitive.
Step 3: Final Answer:
Since the relation R is reflexive, symmetric, and transitive, it is an equivalence relation. The correct option is that the relation is reflexive, symmetric and also transitive.