(a) Let A={1,2,3}, B={4}, f={(1,4),(2,4),(3,4)}. Is f a...

(a) Let $A=\{1,2,3\}, B=\{4\}, f=\{(1,4),(2,4),(3,4)\}$. Is $f$ a function from $A$ to $B$ ? (b) Let $A=\{1\}, B=\{2,3,4\}, f=\{(1,2),(1,3),(1,4)\}$. Is $f$ a function from $A$ to $B$ ? (c) Let $f$ be the relation represented by the following graph. Is $f$ a function from $A$ to $B$ ? (Figure can't copy) (d) Let $C$ be the set of all cars registered in your state, and let $S$ be the set of all sequences of at most 10 letters and digits. Let $L=\{(c, s) \in$ $C \times S \mid$ the license plate number of the car $c$ is $s\}$. Is $L$ a function from $C$ to $S$ ? (e) Let $W$ be the set of all words of English, and let $A$ be the set of all letters of the alphabet. Let $f=\{(w, a) \in W \times A \mid$ the letter $a$ occurs in the word $w\}$, and let $g=\{(w, a) \in W \times A \mid$ the letter $a$ is the first letter of the word $w\}$. Is $f$ a function from $W$ to $A$ ? How about $g$ ? (f) John, Mary, Susan, and Fred go out to dinner and sit at a round table. Let $P=$ (John, Mary, Susan, Fred $\}$, and let $R=\{(p, q) \in P \times P$ ? the person $p$ is sitting immediately to the right of the person $q$}. Is $R$ a function from $P$ to $P$ ?

(a) Let $A=\{1,2,3\}, B=\{4\}, f=\{(1,4),(2,4),(3,4)\}$. Is $f$ a function from $A$ to $B$ ?
(b) Let $A=\{1\}, B=\{2,3,4\}, f=\{(1,2),(1,3),(1,4)\}$. Is $f$ a function from $A$ to $B$ ?
(c) Let $f$ be the relation represented by the following graph. Is $f$ a function from $A$ to $B$ ?
(Figure can't copy)
(d) Let $C$ be the set of all cars registered in your state, and let $S$ be the set of all sequences of at most 10 letters and digits. Let $L=\{(c, s) \in$ $C \times S \mid$ the license plate number of the car $c$ is $s\}$. Is $L$ a function from $C$ to $S$ ?
(e) Let $W$ be the set of all words of English, and let $A$ be the set of all letters of the alphabet. Let $f=\{(w, a) \in W \times A \mid$ the letter $a$ occurs in the word $w\}$, and let $g=\{(w, a) \in W \times A \mid$ the letter $a$ is the first letter of the word $w\}$. Is $f$ a function from $W$ to $A$ ? How about $g$ ?
(f) John, Mary, Susan, and Fred go out to dinner and sit at a round table. Let $P=$ (John, Mary, Susan, Fred $\}$, and let $R=\{(p, q) \in P \times P$ ? the person $p$ is sitting immediately to the right of the person $q$}. Is $R$ a function from $P$ to $P$ ?

Key Concepts

Function Definition

A function is a special type of relation that associates each element of a set (the domain) with exactly one element of another set (the codomain). This means that for every input from the domain, there is one and only one output in the codomain, making the mapping well-defined and unambiguous.

Domain and Codomain

The domain is the set of all inputs for which the function is defined, while the codomain is the set of potential outputs. Understanding these sets is crucial to determining whether a relation qualifies as a function, as every element in the domain must have a corresponding element in the codomain according to the function's rule.

Well-Defined Mapping (Uniqueness Requirement)

A key requirement for a relation to be considered a function is that each element of the domain must be associated with exactly one element of the codomain. If an element from the domain is paired with more than one element in the codomain, the relation fails to satisfy this uniqueness condition and is not a function.

Relation

A relation is any subset of the Cartesian product of two sets, relating elements of one set to elements of another. When assessing whether a relation is a function, one must check if it meets the additional criteria of a function, specifically ensuring that it assigns a unique output to every input.

Graph of a Function and the Vertical Line Test

When a function’s relation is represented graphically, the vertical line test is used to determine if it is indeed a function. In a graph, if any vertical line intersects the graph in more than one point, the relation does not define a function because at least one input value corresponds to multiple outputs.

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