List the ordered pairs in the relation R from A={0,1,2,3,4}...

List the ordered pairs in the relation $R$ from $A=\{0,1,2,3,4\}$ to $B=\{0,1,2,3\},$ where $(a, b) \in R$ if and only if $$ \begin{array}{ll}{\text { a) } a=b .} & {\text { b) } a+b=4} \\ {\text { c) } a>b .} & {\text { d) } a | b} \\ {\text { e) } \operatorname{gcd}(a, b)=1 .} & {\text { f) } \operatorname{lcm}(a, b)=2}\end{array} $$

List the ordered pairs in the relation $R$ from $A=\{0,1,2,3,4\}$ to $B=\{0,1,2,3\},$ where $(a, b) \in R$ if and only if
$$
\begin{array}{ll}{\text { a) } a=b .} & {\text { b) } a+b=4} \\ {\text { c) } a>b .} & {\text { d) } a | b} \\ {\text { e) } \operatorname{gcd}(a, b)=1 .} & {\text { f) } \operatorname{lcm}(a, b)=2}\end{array}
$$

Key Concepts

Cartesian Product

The Cartesian product of two sets is the collection of all ordered pairs where the first element is taken from the first set and the second element from the second set. This concept forms the basis for defining relations as subsets of this product.

Relation

A relation is defined as a subset of the Cartesian product that satisfies a specific condition. It maps elements from one set (the domain) to elements of another set (the codomain) based on given criteria.

Arithmetic Conditions in Relations

Relations can be specified by arithmetic conditions such as equality, inequality, or equations (like a + b = 4). These conditions filter the ordered pairs from the Cartesian product to only include those that satisfy the given numerical relationships.

Divisibility

Divisibility refers to the property where one number divides another exactly, without leaving a remainder. This concept is used in defining relations where one element divides the other.

Greatest Common Divisor (gcd)

The greatest common divisor of two numbers is the largest positive integer that evenly divides both. This concept is crucial when defining relations based on relative primality or shared factors between numbers.

Least Common Multiple (lcm)

The least common multiple of two numbers is the smallest positive integer that is a multiple of both. It helps in defining relations where the combined multiples of the elements meet a specific criterion.

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