Question

Writing: State, in your own words, what is meant by "function" and "relation," and describe how a function differs from a relation. You may use examples to help describe these terms.

Short answer

A 'relation' pairs elements from one set to one or more elements of another set, whereas a 'function' pairs each element from one set to exactly one element of the other set. A function is thus a specific type of relation with a unique pairing.

Step-by-step solution

Understanding a Relation

A relation is a connection between two sets of information, where each element from the first set (known as the domain) can be paired with one or more elements from the second set (known as the range). For example, consider a relation where each person is paired with their favorite type of fruit. One person can like many fruits, hence it's a relation.

Understanding a Function

A function is a specific type of relation where each element from the domain is paired with exactly one element from the range. Using the earlier example, for a function, each person would be paired with only one favorite fruit, not multiple.

Distinguishing Between the Two

The main difference is that in a function, a single element from the domain cannot be associated with more than one element from the range. If a domain element is associated with multiple range elements, this violates the definition of a function and it is simply a relation. For instance, a telephone directory is a function, because each person has a unique number, but a list of people's favorite movies could be a relation, as someone might have several favorites.

Key concepts

Definition of Relation

A relation in mathematics essentially describes a connection or relationship between elements of two sets.

This can be visualized by pairing members of the first set, known as the domain, with elements of the second set, termed the range. Imagine having a set of students and a set of musical instruments. If we associate students with the instruments they can play, we've established a relation. A single student might play the piano, the violin, and the flute, representing multiple pairings within the relation.

Definition of Function

Diving deeper into the characteristics of a function, we find that it's a special relation following a rule: every single input (domain element) corresponds to exactly one output (range element).

Picture a vending machine as a real-life function. You select one specific snack (input) and the machine gives you that snack (output). The key here is exclusivity; you press one button, one snack comes out. Similarly, in mathematics, if you have a set representing people and another representing their Social Security Numbers, the pairing would define a function, because each person has a unique SSN.

Difference Between Function and Relation

To highlight the distinction, consider that while all functions are relations, not all relations are functions.

The unique feature of a function is its one-to-one relationship between the domain and the range. For a set of students and a set of their test scores, if a student can only have one score per test, that's a function. Contrast this with a relation where a student might be paired with multiple potential test scores, symbolizing multiple possibilities - that's not a function anymore because the rule of one-to-one correspondence is broken.

Domain and Range in Sets

The terms domain and range are integral in discussing functions and relations. The domain refers to all the possible inputs, or the set of all starting points, while the range is the set of all possible outputs or endings.

To make sense of this, imagine a school's class-course pairing. The domain is the set of all classes offered, and the range is the set of all courses each class can take. In a function, each class would be assigned to just one course, but in a general relation, a class could have multiple possible courses, illustrating the broader possibilities within the range.