If `A={1,2,3,5}, B={2,4,6,8} and C={4,16,36,39} ` are three sets and R is a relation from A to B and S from B to C defined as
`""_(a)R_(b) iff b=2a` where `a in A, b in B`
`""_(b)S_(c) iff c=b^(2)` where `b in B, c in C`
`:.` Find SoR.

Answer

To find the composition of the relations \( R \) and \( S \), denoted as \( S \circ R \), we will follow these steps:

### Step 1: Identify the relations \( R \) and \( S \)

Given:
- \( R \) is defined as \( a R b \) iff \( b = 2a \) where \( a \in A \) and \( b \in B \).
- \( S \) is defined as \( b S c \) iff \( c = b^2 \) where \( b \in B \) and \( c \in C \).

### Step 2: Determine the pairs in relation \( R \)

We will find all pairs \( (a, b) \) such that \( b = 2a \) for \( a \in A \).

- For \( a = 1 \): \( b = 2 \times 1 = 2 \) (in \( B \))
- For \( a = 2 \): \( b = 2 \times 2 = 4 \) (in \( B \))
- For \( a = 3 \): \( b = 2 \times 3 = 6 \) (in \( B \))
- For \( a = 4 \): \( b = 2 \times 4 = 8 \) (in \( B \))
- For \( a = 5 \): \( b = 2 \times 5 = 10 \) (not in \( B \))

Thus, the pairs in relation \( R \) are:
- \( (1, 2) \)
- \( (2, 4) \)
- \( (3, 6) \)
- \( (4, 8) \)

### Step 3: Determine the pairs in relation \( S \)

Next, we will find all pairs \( (b, c) \) such that \( c = b^2 \) for \( b \in B \).

- For \( b = 2 \): \( c = 2^2 = 4 \) (in \( C \))
- For \( b = 4 \): \( c = 4^2 = 16 \) (in \( C \))
- For \( b = 6 \): \( c = 6^2 = 36 \) (in \( C \))
- For \( b = 8 \): \( c = 8^2 = 64 \) (not in \( C \))

Thus, the pairs in relation \( S \) are:
- \( (2, 4) \)
- \( (4, 16) \)
- \( (6, 36) \)

### Step 4: Find the composition \( S \circ R \)

The composition \( S \circ R \) consists of pairs \( (a, c) \) such that there exists a \( b \) where \( (a, b) \in R \) and \( (b, c) \in S \).

- From \( (1, 2) \) in \( R \) and \( (2, 4) \) in \( S \): \( (1, 4) \)
- From \( (2, 4) \) in \( R \) and \( (4, 16) \) in \( S \): \( (2, 16) \)
- From \( (3, 6) \) in \( R \) and \( (6, 36) \) in \( S \): \( (3, 36) \)
- From \( (4, 8) \) in \( R \): No corresponding \( b \) in \( S \) (since \( 8 \) is not in \( S \))

Thus, the pairs in the composition \( S \circ R \) are:
- \( (1, 4) \)
- \( (2, 16) \)
- \( (3, 36) \)

### Final Result

The composition \( S \circ R \) is:
\[ S \circ R = \{ (1, 4), (2, 16), (3, 36) \} \]

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