Question

Given the relation on \(\mathrm{R}=\\{(\mathrm{a}, \mathrm{b}),(\mathrm{b}, \mathrm{c})\\}\) in the set \(\mathrm{A}=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}\\}\) Then the minimum number of ordered pairs which added to \(R\) make it an equivalence relation is (a) 5 (b) 6 (c) 7 (d) 8

Short answer

To make the given relation $R=\{(a, b), (b, c)\}$ an equivalence relation on set $A=\{a, b, c\}$, we need to add a minimum of 3 ordered pairs for reflexivity, 2 ordered pairs for symmetry, and 1 ordered pair for transitivity, totalling to 6 ordered pairs. Therefore, the correct answer is (b) 6.

Step-by-step solution

Checking Reflexivity

Reflexivity: For every element x in A, the ordered pair (x, x) must be in R. The given relation R = {(a, b), (b, c)}. We can see that none of the self-pairs is present, i.e., (a, a), (b, b), and (c, c) are not in R. So, the relation is not reflexive.

Checking Symmetry

Symmetry: If (x, y) is in R, then (y, x) should also be in R. The given relation R has (a, b) and (b, c) but does not have (b, a) and (c, b). Therefore, this relation is not symmetric.

Checking Transitivity

Transitivity: If (x, y) and (y, z) are in R, then (x, z) should also be in R. The given relation R has (a, b) and (b, c), and in order to be transitive, it should have (a, c) as well. Since (a, c) is not in R, the relation is not transitive.

Minimum ordered pairs for Reflexivity

To make the relation reflexive, we need 3 ordered pairs (a, a), (b, b), and (c, c).

Minimum ordered pairs for Symmetry

To make the relation symmetric, we need 2 ordered pairs (b, a) and (c, b).

Minimum ordered pairs for Transitivity

To make the relation transitive, we need 1 ordered pair (a, c).

Calculating the total minimum ordered pairs

To make the given relation an equivalence relation, we need to add a minimum of 3 (for reflexivity) + 2 (for symmetry) + 1 (for transitivity) = 6 ordered pairs. So, the correct answer is (b) 6.

Key concepts

Reflexivity

The concept of reflexivity is central in defining an equivalence relation. Reflexivity means that every element in a set is related to itself. This means for any element \(x\) in the set \(A\), the pair \((x, x)\) must be included in the relation \(R\).

In our example, the set \(A = \{a, b, c\}\) does not have the pairs \((a, a), (b, b), (c, c)\) in the given relation \(R = \{(a, b), (b, c)\}\). Thus, \(R\) is not reflexive because these pairs are missing.

To turn \(R\) into a reflexive relation, we need to explicitly include these missing self-pairs. This will ensure that each element is related to itself.

**Key Points**:

• Reflexivity requires every element to pair with itself.
• Necessary pairs for reflexivity in example are \((a, a), (b, b), (c, c)\).
• Adding these pairs ensures any relation is reflexive.

Symmetry

Symmetry is another crucial property for creating an equivalence relation. A relation is symmetric if whenever a pair \((x, y)\) is in the relation, the pair \((y, x)\) is also present. In simple terms, if \(x\) is related to \(y\), then \(y\) must be related to \(x\).

In our given relation \(R = \{(a, b), (b, c)\}\), the pairs \((b, a)\) and \((c, b)\) are missing, which shows the relation is not symmetric. These must be added to make \(R\) symmetric.

Ensuring symmetry helps establish reciprocal connections between elements in a set.

**Essentials of Symmetry**:

• Symmetry requires both \((x, y)\) and \((y, x)\) to exist in relation.
• Missing pairs exhibiting symmetry in example are \((b, a), (c, b)\).
• Adding these ensures mutual relations between elements.

Transitivity

Transitivity rounds out the requirements for an equivalence relation. It ensures logical linkage across a series of relations. If a relation is transitive, whenever \((x, y)\) and \((y, z)\) are in the relation, \((x, z)\) must also be.

In the case of our relation \(R = \{(a, b), (b, c)\}\), for it to be transitive, the pair \((a, c)\) needs to be included, as \(b\) is the connecting element between \(a\) and \(c\).

Transitivity is crucial as it links elements indirectly connected through another element.

**Basics of Transitivity**:

• A transitive relation requires \((x, z)\) if \((x, y)\) and \((y, z)\) exist.
• In our example, adding \((a, c)\) achieves transitivity.
• Transitivity helps maintain a coherent linkage among elements.