If R = ((1, 1), (3, 1), (2, 3), (4, 2)), then which of the following represents R2, where R2 is R composite R?

To find the composite relation R2 = R o R, we need to find all pairs (a, c) such that there exists a b for which (a, b) and (b, c) are in R.

The given relation R is: R = {(1, 1), (3, 1), (2, 3), (4, 2)}

Let's find R2 step by step:

1. Start with the first pair in R, (1, 1). We are looking for a pair in R that starts with 1 (the second element of our starting pair). We have two such pairs: (1, 1) and (1, 1). So, from the pair (1, 1), we get two pairs in R2: (1, 1) and (1, 1).

2. Take the second pair in R, (3, 1). We are looking for a pair in R that starts with 1. We have two such pairs: (1, 1) and (1, 1). So, from the pair (3, 1), we get two pairs in R2: (3, 1) and (3, 1).

3. Take the third pair in R, (2, 3). We are looking for a pair in R that starts with 3. But there is no such pair. So, we don't get any pair in R2 from (2, 3).

4. Take the fourth pair in R, (4, 2). We are looking for a pair in R that starts with 2. We have one such pair: (2, 3). So, from the pair (4, 2), we get one pair in R2: (4, 3).

So, the composite relation R2 = R o R is: R2 = {(1, 1), (1, 1), (3, 1), (3, 1), (4, 3)}

To find the composite of a relation R with itself (R2), we need to find all pairs (a, c) such that there exists a b for which (a, b) and (b, c) are in R.

The given relation R is: R = {(1, 1), (3, 1), (2, 3), (4, 2)}

Let's find R2 step by step:

1. Start with the first pair in R: (1, 1). We look for another pair in R that starts with 1. We find (1, 1) again. So, (1, 1) is in R2.

2. Next, consider the pair (3, 1). We look for another pair in R that starts with 1. We find (1, 1). So, (3, 1) is in R2.

3. Now, consider the pair (2, 3). We look for another pair in R that starts with 3. We find (3, 1). So, (2, 1) is in R2.

4. Finally, consider the pair (4, 2). We look for another pair in R that starts with 2. We find (2, 3). So, (4, 3) is in R2.

So, R2 = {(1, 1), (3, 1), (2, 1), (4, 3)}

To find R2, which is R composite R, we need to find all pairs (a, c) such that there exists a b for which (a, b) and (b, c) are in R.

The given relation R is ((1, 1), (3, 1), (2, 3), (4, 2)).

Let's find R2 step by step:

1. Start with the first pair in R, (1, 1). The 'b' value here is 1. We look for other pairs in R where 'a' is also 1. We find (1, 1). So, we have a pair for R2: (1, 1).

2. Next, consider the pair (3, 1). The 'b' value is 1. Looking for other pairs in R where 'a' is 1, we find (1, 1). So, we have another pair for R2: (3, 1).

3. Now, consider the pair (2, 3). The 'b' value is 3. But there are no pairs in R where 'a' is 3. So, this pair does not contribute to R2.

4. Finally, consider the pair (4, 2). The 'b' value is 2. But there are no pairs in R where 'a' is 2. So, this pair also does not contribute to R2.

So, R2 is ((1, 1), (3, 1)).

To find R2, which is R composite R, we need to find all pairs (a, c) such that there exists a b for which (a, b) and (b, c) are in R.

The given relation R is: R = {(1, 1), (3, 1), (2, 3), (4, 2)}

Let's find R2 step by step:

1. Look at the first pair in R, (1, 1). Here, a = 1 and b = 1. Now, we need to find a pair in R where the first element is b = 1. We have such a pair: (1, 1). So, we have a pair for R2: (1, 1).

2. The second pair in R is (3, 1). Here, a = 3 and b = 1. We need to find a pair in R where the first element is b = 1. We have such a pair: (1, 1). So, we have another pair for R2: (3, 1).

3. The third pair in R is (2, 3). Here, a = 2 and b = 3. We need to find a pair in R where the first element is b = 3. But there is no such pair in R. So, there is no pair in R2 corresponding to (2, 3).

4. The fourth pair in R is (4, 2). Here, a = 4 and b = 2. We need to find a pair in R where the first element is b = 2. We have such a pair: (2, 3). So, we have another pair for R2: (4, 3).

So, R2 = {(1, 1), (3, 1), (4, 3)}

The composite of a relation R with itself (denoted as R²) is defined as follows: for all (a, b) and (c, d) in R, if b = c, then (a, d) is in R².

Given R = {(1, 1), (3, 1), (2, 3), (4, 2)}, let's find R².

1. Start with the first pair in R: (1, 1). The second element is 1. Look for other pairs in R where the first element is also 1. We find (1, 1). So, (1, 1) is in R².

2. Next, consider the pair (3, 1). The second element is 1. Look for other pairs in R where the first element is also 1. We find (1, 1). So, (3, 1) is in R².

3. Now, consider the pair (2, 3). The second element is 3. Look for other pairs in R where the first element is also 3. We find none. So, there are no pairs in R² from (2, 3).

4. Finally, consider the pair (4, 2). The second element is 2. Look for other pairs in R where the first element is also 2. We find (2, 3). So, (4, 3) is in R².

Therefore, R² = {(1, 1), (3, 1), (4, 3)}.