Question

Let $A = \{2, 3, 5, 7, 9\}$. Let $R$ be the relation on $A$ defined by $xRy$ if and only if $2x \le 3y$. 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 :

• A. 23
• B. 21
• C. 25
• D. 27

Hint:

For symmetry, ignore diagonal elements (where $x=y$) and check if every pair $(x,y)$ has its reflection $(y,x)$ already in the set.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A relation $R$ on set $A$ is a subset of $A \times A$. For a relation to be symmetric, if $(x, y) \in R$, then $(y, x)$ must also be in $R$.

Step 2: Key Formula or Approach:
We first list all ordered pairs $(x, y)$ such that $x, y \in \{2, 3, 5, 7, 9\}$ and $2x \le 3y$.

Step 3: Detailed Explanation:
Let's find the elements of $R$ for each $x \in A$:
- For $x = 2$, $2(2) = 4 \le 3y \implies y \ge 4/3 \approx 1.33$. Possible $y$: $\{2, 3, 5, 7, 9\}$ (5 pairs).
- For $x = 3$, $2(3) = 6 \le 3y \implies y \ge 2$. Possible $y$: $\{2, 3, 5, 7, 9\}$ (5 pairs).
- For $x = 5$, $2(5) = 10 \le 3y \implies y \ge 3.33$. Possible $y$: $\{5, 7, 9\}$ (3 pairs).
- For $x = 7$, $2(7) = 14 \le 3y \implies y \ge 4.66$. Possible $y$: $\{5, 7, 9\}$ (3 pairs).
- For $x = 9$, $2(9) = 18 \le 3y \implies y \ge 6$. Possible $y$: $\{7, 9\}$ (2 pairs).
Total number of elements $l = 5 + 5 + 3 + 3 + 2 = 18$.
Next, we identify pairs $(x, y) \in R$ whose reverse $(y, x) \notin R$ to find $m$:
Pairs $(x, y) \in R$ with $x<y$: $(2,3), (2,5), (2,7), (2,9), (3,5), (3,7), (3,9), (5,7), (5,9), (7,9)$.
- $(2,3) \in R$ and $(3,2) \in R$ ($2(3) \le 3(2) \implies 6 \le 6$).
- $(2,5) \in R$ but $(5,2) \notin R$ ($10 \not\le 6$). Need to add $(5,2)$.
- $(2,7) \in R$ but $(7,2) \notin R$ ($14 \not\le 6$). Need to add $(7,2)$.
- $(2,9) \in R$ but $(9,2) \notin R$ ($18 \not\le 6$). Need to add $(9,2)$.
- $(3,5) \in R$ but $(5,3) \notin R$ ($10 \not\le 9$). Need to add $(5,3)$.
- $(3,7) \in R$ but $(7,3) \notin R$ ($14 \not\le 9$). Need to add $(7,3)$.
- $(3,9) \in R$ but $(9,3) \notin R$ ($18 \not\le 9$). Need to add $(9,3)$.
- $(5,7) \in R$ and $(7,5) \in R$ ($14 \le 15$).
- $(5,9) \in R$ but $(9,5) \notin R$ ($18 \not\le 15$). Need to add $(9,5)$.
- $(7,9) \in R$ and $(9,7) \in R$ ($18 \le 21$).
The missing symmetric pairs are: $\{(5,2), (7,2), (9,2), (5,3), (7,3), (9,3), (9,5)\}$.
Thus, $m = 7$.

Step 4: Final Answer:
$l + m = 18 + 7 = 25$.