explain this concept---**Textual Information:**
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2.3 Relations
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Let A and B be two non-empty sets. Then, a relation R from A to B is a subset of A × B.
Thus, R is a relation from A to B ⇒ R ⊆ A × B. If R is a relation from a non-empty set A to a non-empty set B and if (a, b) ∈ R, then we write aRb which is read as 'a is related to b by the relation R'. If (a, b) ∉ R, then we write a⋔Rb and it is read as 'a is not related to b by the relation R'.
e.g., If R is a relation between two sets A = {1, 2, 3} and B = {1, 4, 9} defined as "square root of".
Here, 1R1, 2R4, 3R9.
∴ R = {(1, 1), (2, 4), (3, 9)}
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Welcome to our exploration of relations in mathematics. A relation R from set A to set B is formally defined as a subset of the Cartesian product A times B. This means that a relation contains some, but not necessarily all, ordered pairs that can be formed between elements of set A and elements of set B. In this visual representation, we see two sets A and B, with arrows showing which elements are related to each other through relation R.
To understand relations, we first need to understand the Cartesian product. Given sets A equals 1, 2, 3 and B equals 1, 4, 9, the Cartesian product A times B contains all possible ordered pairs where the first element comes from A and the second from B. This gives us 9 ordered pairs in total: (1,1), (1,4), (1,9), (2,1), (2,4), (2,9), (3,1), (3,4), and (3,9). This table representation shows how each element from A pairs with each element from B. Any relation R from A to B must be a subset of this complete Cartesian product.
Relations use specific notation to express relationships between elements. When an ordered pair (a, b) belongs to relation R, we write aRb, which reads as 'a is related to b by relation R'. Conversely, when an ordered pair (a, b) does not belong to relation R, we write a not R b, meaning 'a is not related to b by relation R'. For example, if 2R4, this means the ordered pair (2, 4) is in the relation R, shown by the green arrow. However, if 1 not R 9, this means the ordered pair (1, 9) is not in relation R, indicated by the crossed-out connection in red.
Let's work through a concrete example of the square root relation between sets A equals 1, 2, 3 and B equals 1, 4, 9. In this relation, element a is related to element b if a is the square root of b. So 1R1 because the square root of 1 equals 1. Similarly, 2R4 because the square root of 4 equals 2, and 3R9 because the square root of 9 equals 3. This gives us the relation R equals the set containing ordered pairs (1,1), (2,4), and (3,9). Notice that pairs like (1,4) or (2,9) are not in this relation because the square root of 4 is not 1, and the square root of 9 is not 2.
Relations can be represented in three different ways, each with its own advantages. First, as a set of ordered pairs, which provides mathematical precision and is written as R equals the set containing (1,1), (2,4), and (3,9). Second, as an arrow diagram, which offers visual clarity by showing connections between elements of the two sets with arrows. Third, as a table or matrix, which gives a complete overview by showing all possible pairs and marking which ones belong to the relation with checkmarks. The set notation is precise for mathematical work, arrow diagrams help visualize relationships, and tables provide a systematic way to see all possibilities at once.