Let A = {-2, -1, 0, 1, 2, 3}. Let R be a relation on A defined by xRy if and only if y = max{x, 1}.

Q: Wait, why is $(1, 1)$ included in $R$? The definition says $y = \max\{x, 1\}$.

That is a fantastic observation! The definition is $y = \max\{x, 1\}$. When we plug in $x=1$, we calculate $\max\{1, 1\}$, which is $1$. Therefore, the ordered pair $(1, 1)$ is indeed a valid member of the relation $R$. Always trust the definition over your intuition!

Q: Do I need to add $(1, 1)$ to make it reflexive?

No, you do not! Reflexivity requires that $(a, a) \in R$ for all $a \in A$. Since $(1, 1)$ is already present in our set $R$, the condition is already satisfied for $a=1$. You only need to add the pairs that are currently *missing* from the set to satisfy the condition.

Q: Why is $(2, 1)$ not in $R$?

The relation is defined by the rule $y = \max\{x, 1\}$. If we take $x=2$, then $y = \max\{2, 1\} = 2$. Thus, the pair is $(2, 2)$, not $(2, 1)$. The relation is essentially a function here, mapping each input to a specific output. If the rule doesn't produce $(2, 1)$, it simply isn't in the relation.

Question

Let $A = \{-2, -1, 0, 1, 2, 3\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $y = \max\{x, 1\}$. Let $l$ be the number of elements in $R$. Let $m$ 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

Options:

(A) 12

(B) 11

(C) 13

(D) 14

Sigma Percentile · Accepted Answer

### The Architecture of Relations: A Journey into Set Theory

Welcome, fellow traveler in the realm of mathematics. Today, we are not just solving a problem; we are dissecting the anatomy of a relation. Relations are the fundamental building blocks of functions, logic, and structure in mathematics.

When we define a relation $R$ on a set $A$, we are essentially creating a set of rules that dictate how elements of $A$ 'talk' to each other. Let us embark on this journey to understand the set $A = \{-2, -1, 0, 1, 2, 3\}$ and the rule $y = \max\{x, 1\}$.

### Phase 1: The Mapping of the Set

Imagine you are standing at the edge of a number line. You have six numbers: $A = \{-2, -1, 0, 1, 2, 3\}$. We have a rule, a 'gravity' of sorts, defined by $y = \max\{x, 1\}$.

This rule acts like a magnet. If you are a number less than $1$, the magnet pulls you toward $1$. If you are $1$ or greater, you stay exactly where you are.

Let us map this out systematically:
- For $x = -2$, $y = \max\{-2, 1\} = 1$. This gives us the pair $(-2, 1)$.
- For $x = -1$, $y = \max\{-1, 1\} = 1$, giving us $(-1, 1)$.
- For $x = 0$, $y = \max\{0, 1\} = 1$, giving us $(0, 1)$.

Notice the convergence? All these numbers are being 'mapped' to the number $1$. This is the first cluster of our relation.

Now, consider the numbers $x \ge 1$:
- For $x = 1$, $y = \max\{1, 1\} = 1$, yielding $(1, 1)$.
- For $x = 2$, $y = \max\{2, 1\} = 2$, yielding $(2, 2)$.
- For $x = 3$, $y = \max\{3, 1\} = 3$, yielding $(3, 3)$.

We have successfully mapped the entire set $A$. Our relation $R$ is the set of these six ordered pairs:
$$R = \{(-2, 1), (-1, 1), (0, 1), (1, 1), (2, 2), (3, 3)\}$$

Thus, the number of elements $l$ is simply $6$.

### Phase 2: The Quest for Reflexivity

A relation is 'reflexive' if every element is related to itself. It is the mathematical equivalent of looking in a mirror and seeing yourself. For our set $A$, we need $(a, a) \in R$ for every $a \in \{-2, -1, 0, 1, 2, 3\}$.

Let us check our current list. We have $(1, 1)$, $(2, 2)$, and $(3, 3)$. These are perfect.

However, we are missing $(-2, -2)$, $(-1, -1)$, and $(0, 0)$. To make the relation reflexive, we must forcibly add these three pairs to our set. Therefore, the minimum number of elements to add, $m$, is $3$.

### Phase 3: The Quest for Symmetry

Now, we turn to symmetry. A relation is symmetric if, whenever $(a, b) \in R$, then $(b, a)$ must also be in $R$. Think of it as a two-way street; if you can go from $a$ to $b$, you must be able to return from $b$ to $a$.

Let us examine our original set $R = \{(-2, 1), (-1, 1), (0, 1), (1, 1), (2, 2), (3, 3)\}$:
- For $(-2, 1)$, we need $(1, -2)$. It is not there, so we must add it.
- For $(-1, 1)$, we need $(1, -1)$. It is not there, so we must add it.
- For $(0, 1)$, we need $(1, 0)$. It is not there, so we must add it.

The pairs $(1, 1)$, $(2, 2)$, and $(3, 3)$ are already symmetric because they are their own reverses. We have identified three missing pairs: $(1, -2)$, $(1, -1)$, and $(1, 0)$. Thus, the minimum number of elements to add for symmetry, $n$, is $3$.

### Conclusion: The Final Sum

We have navigated the logic of relations. We found $l = 6$ (the initial size), $m = 3$ (the cost of reflexivity), and $n = 3$ (the cost of symmetry).

The problem asks for the sum $l + m + n$. Calculating this is the final step of our journey:
$$6 + 3 + 3 = 12$$

The final answer is **12**.

Select Answer:

Visualized Solution

Defining Set $A$ and Relation $R$

Evaluating $R$ for $x < 1$

Evaluating $R$ for $x \geq 1$

Calculating $l = ∣ R ∣$

Finding $m$ for Reflexivity

Finding $n$ for Symmetry

Final Calculation: $l + m + n$

The Sigma Insight: Types of Relations

The Architecture of Relations

Phase 1

Phase 2

Phase 3

Conclusion

Similar Questions

Let $A = {2, 3, 5, 7, 9}$ . Let $R$ be the relation on $A$ defined by $x R y$ if and only if $2 x \leq 3 y$ . Let $l$ be the number of elements in $R$ , and $m$ be the minimum number of elements required to be added in $R$ to make it a symmetric relation. Then $l + m$ is equal to :

Let $A = {- 4, - 3, - 2, 0, 1, 3, 4}$ and $R = {(a, b) \in A \times A : b = ∣ a ∣ or b^{2} = a + 1}$ be a relation on $A$ . Then the minimum number of elements, that must be added to the relation $R$ so that it becomes reflexive and symmetric, is

The number of relations on the set $A = {1, 2, 3}$ containing at most 6 elements including $(1, 2)$ , which are reflexive and transitive but not symmetric, is ________

Let $A = {0, 3, 4, 6, 7, 8, 9, 10}$ and $R$ be the relation defined on $A$ such that $R = {(x, y) \in A \times A : x - y is odd positive integer or x - y = 2}$ . The minimum number of elements that must be added to the relation $R$ , so that it is a symmetric relation, is equal to

Let $A = {0, 1, 2, 3, 4, 5}$ . Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $max {x, y} \in {3, 4}$ . Then among the statements $(S 1)$ : The number of elements in $R$ is 18, and $(S 2)$ : The relation $R$ is symmetric but neither reflexive nor transitive

Select Answer:

Visualized Solution

Defining Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A and Relation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R

Evaluating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x<1

Evaluating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x≥1

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } l=∣R∣

Finding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } m for Reflexivity

Finding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n for Symmetry

Final Calculation: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } l+m+n