Question

List the ordered pairs in the relation R from

A = {0, 1, 2, 3, 4} to B = {0, 1, 2, 3}, where (a, b) ∈ R

if and only if

a) a = b. b) a + b = 4.

c) a > b. d) a ∣ b.

e) gcd(a, b) = 1. f ) lcm(a, b) = 2.

Answer 1

Let us list the ordered pairs in the relation $R$ from $A =\{0, 1, 2, 3, 4\}$ to $B = \{0, 1, 2, 3\},$ where $(a, b) ∈ R$ if and only if

a) Let $(a, b) ∈ R$ if and only if $a = b.$ Then $R=\{(0,0),(1,1),(2,2),(3,3)\}.$

b)Let $(a, b) ∈ R$ if and only if $a + b = 4.$ It follows that $R=\{(1,3),(2,2),(3,1),(4,0)\}.$

c) Let $(a, b) ∈ R$ if and only if $a > b.$ Then $R=\{(1,0),(2,1),(2,0),(3,2),(3,1),(3,0),\\(4,3),(4,2),(4,1),(4,0)\}.$

d) Let $(a, b) ∈ R$ if and only if $a ∣ b.$ It follows that $R=\{(1,0),(1,1),(1,2),(1,3),(2,0),(2,2),\\(3,0),(3,3),(4,0)\}.$

e) Let $(a, b) ∈ R$ if and only if $gcd(a, b) = 1.$ Then $R=\{(0,1),(1,0),(1,1),(1,2),(2,1),(1,3),(3,1),\\ (4,1),(2,3),(3,2),(4,3)\}.$

f ) Let $(a, b) ∈ R$ if and only if $lcm(a, b) = 2.$ It follows that $R=\{(2,2),(1,2),(2,1)\}.$