Understanding the Relation R on Natural Numbers

The question defines a relation R on the set of natural numbers, denoted by N. The set of natural numbers N is typically considered to be \(\{1, 2, 3, \dots\}\).

The relation R is given by the set of ordered pairs \((a, b)\) such that two conditions are met:

• Condition 1: \(a = b - 2\)
• Condition 2: \(b > 6\)

For any pair \((a, b)\) to be a member of the relation R (i.e., \((a, b) \in R\)), both of these conditions must be simultaneously satisfied. Also, both \(a\) and \(b\) must be natural numbers because the relation is defined on the set N.

Checking Each Option Against the Relation Conditions

We will now test each given option to determine if the pair \((a, b)\) satisfies both Condition 1 (\(a = b - 2\)) and Condition 2 (\(b > 6\)).

Option 1: Checking (2, 4)

Here, \(a = 2\) and \(b = 4\).

• Checking Condition 1: Is \(a = b - 2\)? Is \(2 = 4 - 2\)? Yes, \(2 = 2\). Condition 1 is satisfied.
• Checking Condition 2: Is \(b > 6\)? Is \(4 > 6\)? No, 4 is not greater than 6. Condition 2 is not satisfied.

Since Condition 2 is not satisfied, the pair \((2, 4)\) is not in the relation R.

Option 2: Checking (3, 8)

Here, \(a = 3\) and \(b = 8\).

• Checking Condition 1: Is \(a = b - 2\)? Is \(3 = 8 - 2\)? Is \(3 = 6\)? No, this is false. Condition 1 is not satisfied.
• Checking Condition 2: Is \(b > 6\)? Is \(8 > 6\)? Yes, 8 is greater than 6. Condition 2 is satisfied.

Since Condition 1 is not satisfied, the pair \((3, 8)\) is not in the relation R.

Option 3: Checking (6, 8)

Here, \(a = 6\) and \(b = 8\).

• Checking Condition 1: Is \(a = b - 2\)? Is \(6 = 8 - 2\)? Is \(6 = 6\)? Yes, this is true. Condition 1 is satisfied.
• Checking Condition 2: Is \(b > 6\)? Is \(8 > 6\)? Yes, 8 is greater than 6. Condition 2 is satisfied.

Since both Condition 1 and Condition 2 are satisfied, the pair \((6, 8)\) is in the relation R.

Option 4: Checking (8, 7)

Here, \(a = 8\) and \(b = 7\).

• Checking Condition 1: Is \(a = b - 2\)? Is \(8 = 7 - 2\)? Is \(8 = 5\)? No, this is false. Condition 1 is not satisfied.
• Checking Condition 2: Is \(b > 6\)? Is \(7 > 6\)? Yes, 7 is greater than 6. Condition 2 is satisfied.

Since Condition 1 is not satisfied, the pair \((8, 7)\) is not in the relation R.

Conclusion

Based on the step-by-step checking of each option against the defining conditions of the relation R, only the pair \((6, 8)\) satisfies both \(a = b - 2\) and \(b > 6\). Therefore, \((6, 8) \in R\).

Relation and Set Theory Concepts Revision Table

Additional Information on Relation Properties

Beyond just determining which pairs are in a relation, we can also analyze the properties of the relation itself. For a relation R on a set A, common properties include:

• Reflexivity: R is reflexive if \((x, x) \in R\) for every element \(x \in A\). For our relation \(R = \{(a, b) \mid a = b - 2, b > 6\}\) on N, for \((a, a)\) to be in R, we'd need \(a = a - 2\) and \(a > 6\). \(a = a - 2\) simplifies to \(0 = -2\), which is never true. So, R is not reflexive.
• Symmetry: R is symmetric if whenever \((x, y) \in R\), it implies \((y, x) \in R\). We found \((6, 8) \in R\). For R to be symmetric, \((8, 6)\) would also need to be in R. For \((8, 6)\) to be in R, we need \(8 = 6 - 2\) (false, as \(8 \ne 4\)) and \(6 > 6\) (false). So, R is not symmetric.
• Transitivity: R is transitive if whenever \((x, y) \in R\) and \((y, z) \in R\), it implies \((x, z) \in R\). If \((a, b) \in R\), then \(a = b - 2\) and \(b > 6\). If \((b, c) \in R\), then \(b = c - 2\) and \(c > 6\). For transitivity, we need to check if \((a, c) \in R\). If \((a, b) \in R\), then \(b = a + 2\). Since \(b > 6\), \(a+2 > 6\), so \(a > 4\). If \((b, c) \in R\), then \(c = b + 2\). Substituting \(b = a+2\), we get \(c = (a+2) + 2 = a + 4\). For \((a, c)\) to be in R, we need \(a = c - 2\) and \(c > 6\). From \(c = a + 4\), we have \(a = c - 4\). The condition for R is \(a = c - 2\). Since \(c - 4 \ne c - 2\), the relation R is not transitive.