Question

Give an example of a relation from \(\\{a, b, c, d\\}\) to \(\\{d, e\\}\) that is not a function.

Short answer

An example of a relation from \{a, b, c, d\} to \{d, e\} that is not a function is: R = \{(a, d), (b, d), (c, d), (d, d), (d, e)\} because the element 'd' from the first set is related to both 'd' and 'e' from the second set.

Step-by-step solution

Identifying the criteria for a function

Firstly, recall the criteria for a relation to be a function: Each element in the domain (source set) should be related to exactly one element in the co-domain (target set). Therefore, if a relation has an element in the domain mapped to more than one element in the co-domain, or an element in domain not related to any element in co-domain, it is not a function.

Formulating the relation

Now, formulate a relation from set \{a, b, c, d\} to set \{d, e\} that doesn't meet the criteria for a function. This can be done by relating one element from set \{a, b, c, d\} to more than one element in set \{d, e\}, or leaving some elements in set \{a, b, c, d\} not related to any elements in set \{d, e\}. An example of such a relation can be: R = \{(a, d), (b, d), (c, d), (d, d), (d, e)\}.

Key concepts

Criteria for a Function

Understanding the criteria for a function is crucial when studying relations and functions in mathematics. A function is a special type of relation that connects members of a domain (the set of all possible inputs) to members of a co-domain (the set of all possible outputs) in a very specific way. To meet the criteria of being a function, each element in the domain must be paired with exactly one element in the co-domain.

This one-to-one relationship means for every input, there is only one output. If an element of the domain is paired with multiple elements of the co-domain, or if an element of the domain has no pair at all, it fails to meet the criteria and is not considered a function. This is a foundational concept in mathematics as it ensures a consistent output for each input, which is essential for defining mathematical processes and calculations.

Domain and Co-domain

The concept of domain and co-domain are integral to functions. The domain is the set of all possible inputs for the function, whereas the co-domain is the set of possible outputs. It's important to note that not every element of the co-domain has to be mapped to by an element of the domain; however, every element of the domain must be linked to an element in the co-domain to meet the definition of a function.

An easy way to remember this is to think of the domain as the starting points and the co-domain as the potential finishing points. This is a distinguishing feature from general relations where elements of the domain may relate to multiple elements in the co-domain or none at all. When identities of these sets are well-defined, it helps to understand the nature of the function and the scope of the analysis that can be performed using the function.

Mapping in Functions

Lastly, mapping in functions refers to how elements from the domain are 'mapped' or linked, to elements in the co-domain. This association is symbolized by arrows or pairs of elements, illustrating which input corresponds to which output. In a function, this mapping is deterministic, meaning each input from the domain is associated with one and only one output in the co-domain.

This unique mapping in functions is what allows us to create graphs and to compute values using functions. In contrast to relations where an element from the domain may have no correlation or multiple correlations in the co-domain, a function's clear-cut mapping schema paves the way for precise computations and predictable behaviour, which is fundamental in various branches of mathematics and its applications.