Question

If \( R \) is the relation "less than" from \( A = \{1,2,3,4,5\} \) to \( B = \{1,4,5\} \), find the set of ordered pairs corresponding to \( R \). Also, define this relation from \( B \) to \( A \).

Hint:

The identity \( \mathbf{x} \times \mathbf{y} = -(\mathbf{y} \times \mathbf{x}) \) helps in simplifying vector cross products.

Answer 1

Step 1: Find ordered pairs for \( R \) from \( A \) to \( B \). Since \( R \) is defined by "less than" (\(<\)), the ordered pairs are: \[ R = \{(1,4), (1,5), (2,4), (2,5), (3,4), (3,5), (4,5)\} \] Step 2: Define the relation from \( B \) to \( A \). For \( R^{-1} \) (inverse relation), the ordered pairs are reversed: \[ R^{-1} = \{(4,1), (5,1), (4,2), (5,2), (4,3), (5,3), (5,4)\} \]