Welcome to our exploration of relations between two sets. A relation between two sets A and B is a collection of ordered pairs where the first element comes from set A and the second element comes from set B. In this example, we have set A containing elements 1 and 2, and set B containing elements x and y. The arrows show a specific relation R that connects 1 to x and 2 to y.
To understand relations, we first need to understand the Cartesian product. The Cartesian product A times B is the set of all possible ordered pairs where the first element comes from set A and the second element comes from set B. For example, if A contains 1 and 2, and B contains x and y, then A times B contains four ordered pairs: 1 comma x, 1 comma y, 2 comma x, and 2 comma y. This table shows all possible combinations.
A relation R from set A to set B is any subset of the Cartesian product A times B. This means we can choose any combination of the ordered pairs to form a relation. For example, relation R1 contains only the pairs 1 comma x and 2 comma y. We could also have a total relation that includes all possible pairs, an empty relation with no pairs, or any other subset. The table shows which pairs are included in relation R1 using checkmarks and crosses.
Relations between sets can be effectively visualized using arrow diagrams. In these diagrams, we represent each set as a box or circle containing its elements as dots. When an ordered pair a comma b belongs to the relation R, we draw an arrow from element a in set A to element b in set B. This visual representation makes it easy to see which elements are related to each other. In our example, the arrows show that 1 is related to x and 2 is related to y.
To summarize what we have learned about relations between two sets: A relation from set A to set B is any subset of their Cartesian product. The Cartesian product gives us all possible ordered pairs. We can represent relations using tables or arrow diagrams to visualize the connections. Relations are fundamental concepts that help us understand how elements from different sets can be connected, forming the basis for more advanced topics like functions.