Question

Let $ A = \{-2, -1, 0, 1, 2, 3\} $. Let $ R $ be a relation on $ A $ defined by $ (x, y) \in R $ if and only if $ |x| \le |y| $. Let $ m $ be the number of reflexive elements in $ R $ and $ n $ be the minimum number of elements required to be added in $ R $ to make it reflexive and symmetric relations, respectively. Then $ l + m + n $ is equal to

• A. 12
• B. 13
• C. 11
• D. 14

Hint:

A relation is reflexive if \( (x, x) \) is in the relation for all elements \( x \) in the set. A relation is symmetric if whenever \( (x, y) \) is in the relation, \( (y, x) \) is also in the relation. To make a relation reflexive, add all missing pairs of the form \( (x, x) \). To make a relation symmetric, for every pair \( (x, y) \) in the relation, if \( (y, x) \) is not already present, add it.

Answer 1

The Correct Option is A

Given set \(A = \{-2, -1, 0, 1, 2, 3\}\) and a relation R on A defined by \(y = \max\{x, 1\}\). We are required to determine \(l\), the cardinality of R; \(m\), the minimum number of elements to add to R to make it reflexive; and \(n\), the minimum number of elements to add to R to make it symmetric. The objective is to compute \(l + m + n\).

Concepts:

A relation R on set A is a subset of \(A \times A\).

A relation R on A is reflexive if \((a, a) \in R\) for all \(a \in A\).

A relation R on A is symmetric if for every \((a, b) \in R\), it implies \((b, a) \in R\).

Solution:

Step 1: Determine R and its size, \(l\).

The relation is defined by \(y = \max\{x, 1\}\) for \(x, y \in A\). Evaluating for each \(x \in A\):

• For \(x = -2\), \(y = \max\{-2, 1\} = 1\). Pair: \((-2, 1)\).
• For \(x = -1\), \(y = \max\{-1, 1\} = 1\). Pair: \((-1, 1)\).
• For \(x = 0\), \(y = \max\{0, 1\} = 1\). Pair: \((0, 1)\).
• For \(x = 1\), \(y = \max\{1, 1\} = 1\). Pair: \((1, 1)\).
• For \(x = 2\), \(y = \max\{2, 1\} = 2\). Pair: \((2, 2)\).
• For \(x = 3\), \(y = \max\{3, 1\} = 3\). Pair: \((3, 3)\).

Thus, \(R = \{(-2, 1), (-1, 1), (0, 1), (1, 1), (2, 2), (3, 3)\}\).

The cardinality of R is \(l\).

\[l = 6\]

Step 2: Determine \(m\), the count of elements to add for reflexivity.

For R to be reflexive on A, it must include all pairs \((a, a)\) for \(a \in A\).

The complete set of reflexive pairs is \(R_{\text{reflexive\_pairs}} = \{(-2, -2), (-1, -1), (0, 0), (1, 1), (2, 2), (3, 3)\}\).

R already contains \((1, 1), (2, 2), (3, 3)\). The missing pairs are \(\{(-2, -2), (-1, -1), (0, 0)\}\).

The minimum number of elements to add is:

\[m = 3\]

Step 3: Determine \(n\), the count of elements to add for symmetry.

For R to be symmetric, if \((a, b) \in R\), then \((b, a)\) must also be in R.

• \((-2, 1) \in R\) requires \((1, -2)\).
• \((-1, 1) \in R\) requires \((1, -1)\).
• \((0, 1) \in R\) requires \((1, 0)\).
• \((1, 1), (2, 2), (3, 3)\) are already symmetric with themselves.

The pairs to be added for symmetry are \(\{(1, -2), (1, -1), (1, 0)\}\).

The minimum number of elements to add is:

\[n = 3\]

Step 4: Compute \(l + m + n\).

Using \(l=6\), \(m=3\), and \(n=3\):

\[l + m + n = 6 + 3 + 3 = 12\]

The value of \(l + m + n\) is 12.