Relations

In the realm of mathematics, Relations serve as the bridges that connect elements from diverse sets. These connections can be depicted through various methods including sets of ordered pairs, matrices, or even graphical illustrations. Specific types of relations, such as equivalence and ordering relations, exhibit unique properties that are widely applied across multiple mathematical disciplines.

Determining if the Relation is a Function

Given the relation \( R = \{(1, 3), (2, 3), (1, 4), (2, 5)\} \), determine if the relation is a function.

Finding the Domain and Range of the Relation

Find the domain and range of the relation \( R = \{ (1,2), (3,4), (5,6), (7,8) \} \).

Finding the Inverse of the Relation

Let the relation \( R \) be defined on the set of real numbers by \( R = \{(x, y) | x = 2y + 3\} \). Find the inverse of the relation \( R \).

Determining if Surjective (Onto)

Let's consider a function \(f: A \rightarrow B\) where \(A = \{1, 2, 3, 4\}\) and \(B = \{5, 6, 7, 8\}\). The function is defined as \(f(x) = x + 4\). Is this function surjective (onto)?

Determining if Bijective (One-to-One)

Given the relation \( R = \{(1, 2), (2, 3), (3, 4), (4, 5)\} \), determine if it is bijective.