Question

Let \(R\) be the relation on the set of real numbers defined by \(\mathrm{a} \mathrm{R} \mathrm{b}\). If \(|\mathrm{a}-\mathrm{b}| \leq 1\). Then \(\mathrm{R}\) is (1) Reflexive and symmetric (2) Symmetric only (3) Transitive only (4) Anti-symmetric only

Short answer

(1) Reflexive and symmetric

Step-by-step solution

Understand the relation

The given relation is defined by \( a R b \) if \( |a - b| \leq 1 \). This means the absolute difference between any two numbers in this relation must be less than or equal to 1.

Check Reflexiveness

A relation R is reflexive if every element is related to itself. For any real number \(a\), \(|a - a| = 0\leq 1 \). Hence, the relation is reflexive.

Check Symmetric Property

A relation R is symmetric if whenever \(aRb\), then \(bRa\). Given that \(aRb\) implies \(|a - b| \leq 1\), it follows that \(|b - a| = |a - b| \leq 1 \), which satisfies \(bRa\). Therefore, the relation is symmetric.

Check Transitivity

A relation R is transitive if whenever \(aRb\) and \(bRc\), then \(aRc\). Assume \(aRb\) and \(bRc\), which means \( |a - b| \leq 1 \) and \( |b - c| \leq 1 \). However, this does not necessarily imply \( |a - c| \leq 1\). For example, if \(a = 0\), \(b = 0.5\), and \(c = 1.5\), then \(|a - b| = 0.5 \leq 1\) and \(|b - c| =1 \leq 1\), but \(|a - c| = 1.5 \ot\leq 1 \). Therefore, the relation is not transitive.

Check Anti-Symmetry

A relation R is anti-symmetric if \(aRb\) and \(bRa\) imply \(a = b\). Given that \(aRb\) and \(bRa\) imply \(|a - b| \leq 1\), this does not necessarily mean \(a = b\). Therefore, the relation is not anti-symmetric.

Conclusion

Since the relation \(R\) is both reflexive and symmetric but not transitive or anti-symmetric, the correct answer is (1) Reflexive and symmetric.

Key concepts

reflexive relation

A relation is termed reflexive if every element is related to itself. This means for a set of real numbers, if we consider an element 'a', it should be such that the relation holds true when related to itself. This is mathematically represented as \(aRa\). For our given relation, defined by \(aRb\) if \(|a - b| \leq 1\), clearly for any real number 'a' \(|a - a| = 0 \leq 1\), meaning every element is connected to itself. Recognizing this property helps validate that the relation is indeed reflexive by checking against all possible elements in the set. Breaking the concept into simpler terms, you can think of it as each object 'acknowledging' itself as part of the group.

symmetric relation

A relation is symmetric if \(aRb\) directly implies \(bRa\). In simpler terms, if an element 'a' is related to an element 'b', then 'b' must also be related to 'a'. This can be visualized as a two-way street connecting the elements. For the given relation \(aRb\) if \(|a - b| \leq 1\), the symmetry is evident because the absolute value function \(|a - b|\) is the same as \(|b - a|\). This property assures that the conditions for 'a' being related to 'b' are identical to 'b' being related to 'a'. It's important to see symmetry as a mutual relationship where if one acknowledges the other, the acknowledgment is reciprocated.

transitive relation

A relation is transitive if whenever \(aRb\) and \(bRc\) hold true, then \(aRc\) must also hold. This seems like a logical chain reaction: if 'a' is related to 'b' and 'b' is related to 'c', then 'a' should be directly related to 'c'. Let's test this on our relation \(|a - b| \leq 1\). Assume \(aRb\) and \(bRc\) are true, meaning \(|a - b| \leq 1\) and \(|b - c| \leq 1\). However, this does not ensure the transitivity since it’s plausible to find cases where even if the individual relations hold, the combined relation \(|a - c|\) might still exceed 1. For example, if \(a = 0\), \(b = 0.5\), and \(c = 1.5\), we have \(|a - b| = 0.5\) and \(|b - c| = 1\) which satisfy the given condition individually, but \(|a - c| = 1.5\) fails the condition, revealing that the relation is not transitive.

anti-symmetric relation

Anti-symmetry in a relation specifies that if \(aRb\) and \(bRa\) hold true, then \(a\) must be equal to \(b\). It implies that mutual relations are strictly possible only when both elements are identical. For our defined relation, \(aRb\) and \(bRa\) translates to \(|a - b| \leq 1\) and \(|b - a| \leq 1\), which does not necessarily conclude that \(a = b\). This gap is where anti-symmetry does not hold since there exist instances where two different real numbers might satisfy the given condition mutually without being the same number. Visualize anti-symmetry as a strict one-way sign that only permits equality if a two-way relationship exists.