Answer to Question #209910 in Discrete Mathematics for Sach

Question

Write down the relation R on the S = {1, 2, 3, 4, 5, 6} by any one of the methods: set of ordered pairs, 

directed graphs, matrix notation. R is given as follows. 

(a) R = {(a, b) | a = b }

(b) R = {(a, b) | a ≠ b}

(c) R = {(a, b) | a < b}

(d) R = {(a, b) | a ≤ b}

(e) R = {(a, b) | a > b}

(f) R = {(a, b) | a ≥ b}

(g) R = {(a, b) | a = b + 1}

(h) R = {(a, b) | a + b ≤ 3}

(i) R = {(a, b) | a divides b}

Answer 1

Solution:

S = {1, 2, 3, 4, 5, 6}

(a) R = {(a, b) | a = b }

R = { (1,1),(2,2),(3,3),(4,4)(5,5),(6,6)}

(b) R = {(a, b) | a ≠ b}

R = { (1,2),(2,3),(3,4)(4,5),(5,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6),(2,1),(3,2),(4,3),(5,4),(6,5),(3,1),(4,1),(5,1),(6,1),(4,2),(5,2),(6,2),(5,3),(6,3),(5,4),(6,4),(6,5) }

(c) R = {(a, b) | a < b}

R = {(1,2),(2,3),(3,4)(4,5),(5,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)}

(d) R = {(a, b) | a ≤ b}

R = {(1,2),(2,3),(3,4)(4,5),(5,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6),(1,1),(2,2),(3,3),(4,4)(5,5),(6,6)}

(e) R = {(a, b) | a > b}

R ={(2,1),(3,2),(4,3),(5,4),(6,5),(3,1),(4,1),(5,1),(6,1),(4,2),(5,2),(6,2),(5,3),(6,3),(5,4),(6,4),(6,5) }

(f) R = {(a, b) | a ≥ b}

R = {(2,1),(3,2),(4,3),(5,4),(6,5),(3,1),(4,1),(5,1),(6,1),(4,2),(5,2),(6,2),(5,3),(6,3),(5,4),(6,4),(6,5),(1,1),(2,2),(3,3),(4,4)(5,5),(6,6) }

(g) R = {(a, b) | a = b + 1}

R = {(2,1),(3,2),(4,3),(5,4),(6,5)}

(h) R = {(a, b) | a + b ≤ 3}

R = {(1,1),(2,1)}

(i) R = {(a, b) | a divides b}

R = {(1,1),(2,2),(3,3),(4,4)(5,5),(6,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,6),(3,6)}