Use elements in the order given to determine rows and columns of the matrix.

R on {1,2,3,4} where aRb means |a-b| <1.

R on {1,2,3,4,6,12} where aRb means a|b.

Adam, Nat, Martin, Tony, or whoever ~ Don't bother changing names. It's obvious you're all the same.

To quote one of our very good math and science tutors: “You will find here at Jiskha that long series of questions, posted with no evidence of effort or thought by the person posting, will not be answered. We will gladly respond to your future questions in which your thoughts are included.”

Adam, Nat, Martin, Tony,

If you have no clue what the questions are asking, it's high time to do some revision before the finals are here.
Show some efforts and say where you need help. Even answers at the end of a book are intended for comparison with your own answers.

To determine the rows and columns of the matrix for each relation, we need to consider each element in the given sets and check how they relate to each other based on the given definitions of the relations.

Let's start with the relation R on {1, 2, 3, 4} where aRb means |a - b| < 1.

Step 1: Determine Rows:
To determine the rows of the matrix, we consider each element in the set {1, 2, 3, 4} and pair them up with every other element in the set. If the absolute difference between the two elements is less than 1, we assign a "1" in the matrix; otherwise, we assign a "0".

Rows:
1R1, 1R2, 1R3, 1R4
2R1, 2R2, 2R3, 2R4
3R1, 3R2, 3R3, 3R4
4R1, 4R2, 4R3, 4R4

Step 2: Determine Columns:
To determine the columns of the matrix, we need to consider the pairs of elements in the set {1, 2, 3, 4} and check the condition |a - b| < 1 for each pair. If the condition is satisfied, we assign a "1" in the matrix; otherwise, we assign a "0".

Columns:
1R1, 1R2, 1R3, 1R4
2R1, 2R2, 2R3, 2R4
3R1, 3R2, 3R3, 3R4
4R1, 4R2, 4R3, 4R4

Now let's move on to the relation R on {1, 2, 3, 4, 6, 12} where aRb means a|b.

Step 1: Determine Rows:
To determine the rows of the matrix, we consider each element in the set {1, 2, 3, 4, 6, 12} and pair them up with every other element in the set. If a divides b (a|b), we assign a "1" in the matrix; otherwise, we assign a "0".

Rows:
1R1, 1R2, 1R3, 1R4, 1R6, 1R12
2R1, 2R2, 2R3, 2R4, 2R6, 2R12
3R1, 3R2, 3R3, 3R4, 3R6, 3R12
4R1, 4R2, 4R3, 4R4, 4R6, 4R12
6R1, 6R2, 6R3, 6R4, 6R6, 6R12
12R1, 12R2, 12R3, 12R4, 12R6, 12R12

Step 2: Determine Columns:
To determine the columns of the matrix, we need to consider the pairs of elements in the set {1, 2, 3, 4, 6, 12} and check the condition a|b for each pair. If the condition is satisfied, we assign a "1" in the matrix; otherwise, we assign a "0".

Columns:
1R1, 1R2, 1R3, 1R4, 1R6, 1R12
2R1, 2R2, 2R3, 2R4, 2R6, 2R12
3R1, 3R2, 3R3, 3R4, 3R6, 3R12
4R1, 4R2, 4R3, 4R4, 4R6, 4R12
6R1, 6R2, 6R3, 6R4, 6R6, 6R12
12R1, 12R2, 12R3, 12R4, 12R6, 12R12

By following these steps, you can determine the rows and columns of the matrix for each given relation.