Let A= 1,2,3,4 and R be a relation on the set A A defined by R= ((a,b),(c,d)): 2a+3b=4c+5d Then the number of elements in R is

Given: A= 1,2,3,4 and R= ((a,b),(c,d)): 2a+3b=4c+5d Find: The number of elements in R. From the solution working, first list attainable values of 4c+5d for c,d A. The valid values taken forward are 9,13,14,17,18 Now solve 2a+3b=k for these values. For k=9: 2a+3b=9 (a,b)=(3,1),(1,3) For k=13: 2a+3b=13 (a,b)=(2,3),(4,1) For k=14: 2a+3b=14 (a,b)=(1,4),(4,2) For k=17: 2a+3b=17 (a,b)=(4,3) For k=18: 2a+3b=18 (a,b)=(3,4) Match these with the corresponding (c,d) values stated in the extracted solution: aligned 9 &:(c,d)=(1,1) 2 elements 13 &:(c,d)=(2,1) 2 elements 14 &:(c,d)=(1,2) 2 elements 17 &:(c,d)=(3,1) 1 element 18 &:(c,d)=(2,2) 1 element aligned The extracted solution then states that only the first three cases contribute and concludes |R|=2+2+2=6 Therefore, the number of elements in R is 6.

Let A= 1,2,3,4 and R be a relation on the set A A defined by R= (… — Solution | JEE Mathematics

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Let $A=\{1,2,3,4\}$ and $R$ be a relation on the set $A\times A$ defined by

R=\{((a,b),(c,d)):\;2a+3b=4c+5d\}

Then the number of elements in $R$ is

Answer

Correct answer:6

Step-by-step solution

Standard Method

Given: $A=\{1,2,3,4\}$ and

R=\{((a,b),(c,d)):\;2a+3b=4c+5d\}

Find: The number of elements in $R$ .

From the solution working, first list attainable values of $4c+5d$ for $c,d\in A$ . The valid values taken forward are

\{9,13,14,17,18\}

Now solve $2a+3b=k$ for these values.

For $k=9$ :

2a+3b=9 \Rightarrow (a,b)=(3,1),(1,3)

For $k=13$ :

2a+3b=13 \Rightarrow (a,b)=(2,3),(4,1)

For $k=14$ :

2a+3b=14 \Rightarrow (a,b)=(1,4),(4,2)

For $k=17$ :

2a+3b=17 \Rightarrow (a,b)=(4,3)

For $k=18$ :

2a+3b=18 \Rightarrow (a,b)=(3,4)

Counting from the extracted working

Match these with the corresponding $(c,d)$ values stated in the extracted solution:

\begin{aligned} 9 &:(c,d)=(1,1) \Rightarrow 2 \text{ elements}\\ 13 &:(c,d)=(2,1) \Rightarrow 2 \text{ elements}\\ 14 &:(c,d)=(1,2) \Rightarrow 2 \text{ elements}\\ 17 &:(c,d)=(3,1) \Rightarrow 1 \text{ element}\\ 18 &:(c,d)=(2,2) \Rightarrow 1 \text{ element} \end{aligned}

The extracted solution then states that only the first three cases contribute and concludes

|R|=2+2+2=6

Therefore, the number of elements in $R$ is $6$ .

Common mistakes

Counting all possible values of $4c+5d$ without checking whether the same value is attainable by $2a+3b$ . This is wrong because a relation element requires equality on both sides. First identify common attainable values, then count matching ordered pairs.
Treating each common value as contributing only one relation element. This is wrong because one RHS pair can match multiple LHS pairs, or vice versa. Count the number of $(a,b)$ solutions and multiply by the number of corresponding $(c,d)$ solutions for each common value.
Forgetting that elements of $R$ are ordered pairs of ordered pairs, namely $((a,b),(c,d))$ . This is wrong because counting only $(a,b)$ or only $(c,d)$ undercounts the relation. Each valid matching combination gives a distinct element of $R$ .

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