Question

Decide whether the set of ordered pairs represents a function from \(A\) to \(B\). \(A=\\{0,1,2,3\\}\) and \(B=\\{-2,-1,0,1,2\\}\) Give reasons for your answers. \(\\{(0,0),(1,0),(2,0),(3,0)\\}\)

Short answer

Yes, the set of ordered pairs \{(0,0),(1,0),(2,0),(3,0)\} does represent a function from \(A\) to \(B\) because each input from set \(A\) is associated with exactly one output in set \(B\).

Step-by-step solution

Clarify the sets A and B

We have two sets here. Set \(A=\{0,1,2,3\}\) is our set of inputs, and set \(B=\{-2,-1,0,1,2\}\) is our set of possible outputs. It's important to note that a function from \(A\) to \(B\) means every element in \(A\) should be associated with exactly one element in \(B\).

Examine the set of ordered pairs

The ordered pairs are given as \(\{(0,0),(1,0),(2,0),(3,0)\}\). These pairs represent the mapping from set \(A\) to set \(B\). Each pair is in the form (input, output). It means the first element of the pair is from set \(A\) (the input) and the second element is from set \(B\) (the output).

Check if the ordered pairs constitutes a function

A function is a relation that assigns each element of a set \(A\) to exactly one element in set \(B\). Looking at our ordered pairs, it's clear that every element 0,1,2,3 in set \(A\) maps to a single element 0 in set \(B\). Hence each input from set \(A\) has exactly one output in set \(B\).

Key concepts

Function Definition

A function in mathematics is a special type of relation that connects elements of one set, known as the domain, to elements of another set, often referred to as the range. This connection is done in such a way that each element in the domain is associated with exactly one element in the range.

Think of a function as a machine where you throw in an input (an element from the domain), and the machine spits out exactly one output (an element from the range). In the context of the given exercise with sets A and B, our 'machine' would take numbers from set A and gives us a corresponding number from set B. The 'function' is the consistent rule that the 'machine' uses to produce the output from the input.

Input-Output Mapping

Input-output mapping in a function illustrates the specific rule that pairs each input with one output. If we think of a function as a party where every guest from the domain (input set A) must dance with exactly one partner from the range (output set B), then the set of ordered pairs represents the dance partners each has chosen.

In our example with sets A and B, the ordered pairs \(\{(0,0),(1,0),(2,0),(3,0)\}\) show that each guest from set A has chosen '0' from set B to dance with. No guest is alone, and no guest is trying to dance with more than one partner, which perfectly adheres to the function's 'one input to one output' rule, confirming that the set of ordered pairs indeed represents a function.

Set Theory in Mathematics

Set theory is the mathematical study of collections of objects, which can be anything from numbers to symbols. In set theory, we use curly braces \(\{\}\) to denote a set, and its elements can be listed inside these braces, like A and B in our exercise.

In the realm of functions, set theory becomes especially important as it provides the language to talk about domains, ranges, and relations. Domains and ranges are sets, and functions are specific types of relations between these sets that meet the criteria of linking each element from the domain to one and only one element in the range.

When analyzing the problem at hand, set theory helps in understanding that the ordered pairs are subsets of the product of sets A and B, where product denotes all possible combinations of elements from the two sets. However, for a subset of this product to be a function, it must assign exactly one element of B to each element of A, as our example set clearly does.