1. Let S be the Cartesian coordinate plane R × R and define a relation R on S by (a, b) R (c, d ) iff a = c. Verify that R is an equivalence relation and describe a typical equivalence class E(a,b) 2....

1. Let S be the Cartesian coordinate plane R × R and define a relation R on S by (a, b) R (c, d ) iff a = c. Verify that R is an equivalence relation and describe a typical equivalence class E(a,b)

2. Let S be the Cartesian coordinate plane R × R and define a relation R on S by (a, b) R (c, d ) iff a + d = b + c. Verify that R is an equivalence relation and describe the equivalence class E(7,3)

3. Let S be the Cartesian coordinate plane R × R and define the equivalence relation R on S by (a, b) R (c, d ) iff a + 2b = c + 2d.

(a) Find the partition p determined by R by describing the pieces in p .

(b) Describe the piece of the partition that contains the point (5, 3).