Dear Student after discussion of Sets and related properties of Sets, We shall now discuss the Relations and Functions.
Ordered Pairs
Suppose, for instance, that the set $A= \{a, b, c, d\}$ of distinct elements, and suppose that we want to consider its elements in the order $cb d a$. We may have the sets $\{c\}, \{c, b\}, \{c, b, d\}, \{c, b, d, a\}$. We can go on then to consider the set (or collection) ,
$ $$\mathcal{C}$$ = \{\{a,b, c, d\}, \{b,c\}, \{b, c, d\}, \{c\}\}$
If $A =\{a, b\}$ and if, in the desired order, $a$ comes first, then $ $$\mathcal{C}$$= \{\{a\}, \{a, b\}\}$; if, however, $b$ comes first, then $ $$\mathcal{C}$$= \{\{b\}, \{a, b\}\}$
The ordered pair of $a$ and $b$, with first coordinate $a$ and second coordinate $b$, is the set $(a, b)$ defined by $(a, b) = \{\{a\}, \{a, b\}\}$.
Note that
If $(a, b)$ and $(x, y)$ are ordered pairs and if $(a, b) = (x, y)$, then $a = x$ and $b = y$.
Cartesian product of $A$ and $B$ is characterized by $A \times B= \{ x \mid x =(a, b)$ for some $a \in A$ and for some $b \in B\}$.
The Cartesian product of two sets is a set of ordered pairs (that is, a set each of whose elements is an ordered pair), and the same is true of every subset of a Cartesian product.
For example, if $A = \{1, 2\}$ and $B = \{2, 3, 4\}$, then $A \times B = \{(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4)\}$ and $B \times A = \{(2, 1), (2, 2), (3, 1), (3, 2), (4, 1), (4, 2)\}$
Therefore , for instance, we have $(1, 2) \in A\times B$ and $(1, 2) \notin B \times A$
If $A, B, C$, and $D$ are sets, then
(i) $(A \cup B)\times C = (A \times C) \cup (B\times C)$,
(ii) $(A \cap B) \times (C \cap D) = (A \times C) \cap (B\times D)$
(iii) $(A- B) \times C = (A\times C) - (B\times C)$
If either $A = \varnothing $ or $B = \varnothing $, then $A\times B =\varnothing $, and conversely. If $A \subset C, B \subset D$, then $A \times B \subset C \times D$, and (provided $A \times B \neq \varnothing$) conversely.
RELATIONS:
A set $R$ is a relation if each element of $R$ is an ordered pair; this means, if $z \in R$, then there exist $x$ and $y$ so that $z = (x, y)$ . If $R$ is a relation, it is sometimes convenient to express the fact that $(x,y)\in R$ by writing $x R y$ For example: let $A$ be any set, and let $R$ be the set of all those pairs $(x, y)$ in $ A \times A $ for which $x = y$. The relation $R$ is just the relation of equality between elements of $A$; if $x$ and $y$ are in $A$, then $x R y$ means the same as $x = y$.
Domain and Range of $R$: We define dom $R = \{ x \mid $ for some $y (x R y)\}$ and Range $R = \{ y \mid $ for some $x (x R y)\}$
Both the domain and the range of $\varnothing$ are equal to $\varnothing$. If $R = A\times B$, then dom $R = A$ and ran $R = B$. If $R$ is equality in $A$, then dom $R = $ range $R = A$
If $ R $ is belonging, between $A$ and $ $$\mathcal{P}$$ (A) $, then dom $R = A$ and ran $R = $$\mathcal{P}$$ (A) - \{\varnothing\}$
If $R$ is a relation included in a Cartesian product $A\times B$ (so that dom $R \subset A$ and ran $R \subset B$), it is sometimes convenient to say that $R$ is a relation from $A$ to $B$; instead of a relation from $A$ to $A$ we may speak of a relation in $A$. A relation $R$ in $A$ is reflexive if $x R x$ for every $x$ in $A$; it is symmetric if $x R y$ implies that $y R x$; and it is transitive if $x R y$ and $y R z$ imply that $x R z$.
FUNCTIONS
If $A$ and $B$ are sets, a, function from (or on) $A$ to (or into) $B$ is a relation $f$ such that dom $f = A$ and such that for each $x$ in $A$ there is a unique element $y$ in $B$ with $(x, y) \in f $. The uniqueness condition can be formulated explicitly as follows: if $(x, y) \in f $ and $(x, z) \in f $, then $y = z$. For each $x$ in $A$, the unique $y$ in $B$ such that $(x, y) \in f $ is denoted by $f(x)$. From now on, if $f$ is a function, we shall write $f(x) = y$ instead of $(x, y) \in f $ or $x f y$. The element $y$ is called the value that the function $f$ assumes (or takes on) at the argument $x$; equivalently we may say that $f$ sends or maps or transforms $x$ onto $y$. The words map or mapping, transformation, correspondence, and operator are among some of the many that are sometimes used as synonyms for function. The symbol $f:A \rightarrow B $ is sometimes used as an abbreviation for "$f$ is a function from $A$ to $B$." The set of all functions from $A$ to $B$ is a subset of the power set $ $$\mathcal{P}$$(A \times B)$; it will be denoted by $B^{A}$.
The domain of a function $f$ from $A$ into $B$ is, by definition, equal to $A$, but its range need not be equal to $B$; the range consists of those elements $y$ of $B$ for which there exists an $x$ in $A$ such that $f(x) = y$. If the range of $f$ is equal to $B$, we say that $f$ maps $A$ onto $B$. If $E$ is a subset of $A$, we may want to consider the set of all those elements $y$ of $B$ for which there exists an $x$ in the subset $E$ such that $f(x) = y$. This subset of $B$ is called the image of $E$ under $f$ and is frequently denoted by $f(E)$. (this means we consider the set of values of $f$ at the elements of $E$ ). Note that the image of $A$ itself is the range of $f$; the "onto" character of $f$ can be expressed by writing $f(A) = B$.
If $A$ is a subset of a set $B$, the function $f$ defined by $f(x) = x$ for each $x$ in $A$ is called the inclusion map (or the embedding, or the injection) of $A$ into $B$. The inclusion map of $A$ into $A$ is called the identity map on $A$.
If $f$ is a function from $B$ to $C$, say, and if $A$ is a subset of $B$, then there is a natural way of constructing a function $g$ from $A$ to $C$; define $g(x)$ to be equal to $f(x)$ for each $x$ in $A$. The function $g$ is called the restriction of $f$ to $A$, and $f$ is called an extension of $g$ to $B$; The definition of restriction can be expressed by writing $(f \mid A)(x) = f(x)$ for each $x$ in $A$; observe also that ran $(f \mid A) = f(x) $.
A function that always maps distinct elements onto distinct elements is called one-to-one (usually a one-to-one correspondence).
INVERSE FUNCTION
Associated with every function $f$, from $A$ to $B$, say, there is a function from $ $$\mathcal{P}$$ (A) $ to $ $$\mathcal{P}$$ (B)$, namely the function (frequently called $f$ also) that assigns to each subset $E$ of $A$ the image subset $f(E)$ of $B$.
Given a function $f$ from $A$ to $B$, let $f^{-1}$, the inverse of $f$, be the function from $ $$\mathcal{P}$$ (B)$ to $ $$\mathcal{P}$$ (A) $ such that if $D \subset B$ , then $f^{-1}(D) = \{x \in A \mid f(x) \in D \}$.
In words: $f^{-1}(D)$ consists of exactly those elements of $A$ that $f$ maps into $D$; the set $f^{-1}(D)$ is called the inverse image of $D$ under $f$. A necessary and sufficient condition that $f$ map $A$ onto $B$ is that the inverse image under $f$ of each non-empty subset of $B$ be a non-empty subset of $A$.
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Ordered Pairs
Suppose, for instance, that the set $A= \{a, b, c, d\}$ of distinct elements, and suppose that we want to consider its elements in the order $cb d a$. We may have the sets $\{c\}, \{c, b\}, \{c, b, d\}, \{c, b, d, a\}$. We can go on then to consider the set (or collection) ,
$ $$\mathcal{C}$$ = \{\{a,b, c, d\}, \{b,c\}, \{b, c, d\}, \{c\}\}$
If $A =\{a, b\}$ and if, in the desired order, $a$ comes first, then $ $$\mathcal{C}$$= \{\{a\}, \{a, b\}\}$; if, however, $b$ comes first, then $ $$\mathcal{C}$$= \{\{b\}, \{a, b\}\}$
The ordered pair of $a$ and $b$, with first coordinate $a$ and second coordinate $b$, is the set $(a, b)$ defined by $(a, b) = \{\{a\}, \{a, b\}\}$.
Note that
If $(a, b)$ and $(x, y)$ are ordered pairs and if $(a, b) = (x, y)$, then $a = x$ and $b = y$.
Cartesian product of $A$ and $B$ is characterized by $A \times B= \{ x \mid x =(a, b)$ for some $a \in A$ and for some $b \in B\}$.
The Cartesian product of two sets is a set of ordered pairs (that is, a set each of whose elements is an ordered pair), and the same is true of every subset of a Cartesian product.
For example, if $A = \{1, 2\}$ and $B = \{2, 3, 4\}$, then $A \times B = \{(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4)\}$ and $B \times A = \{(2, 1), (2, 2), (3, 1), (3, 2), (4, 1), (4, 2)\}$
Therefore , for instance, we have $(1, 2) \in A\times B$ and $(1, 2) \notin B \times A$
If $A, B, C$, and $D$ are sets, then
(i) $(A \cup B)\times C = (A \times C) \cup (B\times C)$,
(ii) $(A \cap B) \times (C \cap D) = (A \times C) \cap (B\times D)$
(iii) $(A- B) \times C = (A\times C) - (B\times C)$
If either $A = \varnothing $ or $B = \varnothing $, then $A\times B =\varnothing $, and conversely. If $A \subset C, B \subset D$, then $A \times B \subset C \times D$, and (provided $A \times B \neq \varnothing$) conversely.
RELATIONS:
A set $R$ is a relation if each element of $R$ is an ordered pair; this means, if $z \in R$, then there exist $x$ and $y$ so that $z = (x, y)$ . If $R$ is a relation, it is sometimes convenient to express the fact that $(x,y)\in R$ by writing $x R y$ For example: let $A$ be any set, and let $R$ be the set of all those pairs $(x, y)$ in $ A \times A $ for which $x = y$. The relation $R$ is just the relation of equality between elements of $A$; if $x$ and $y$ are in $A$, then $x R y$ means the same as $x = y$.
Domain and Range of $R$: We define dom $R = \{ x \mid $ for some $y (x R y)\}$ and Range $R = \{ y \mid $ for some $x (x R y)\}$
Both the domain and the range of $\varnothing$ are equal to $\varnothing$. If $R = A\times B$, then dom $R = A$ and ran $R = B$. If $R$ is equality in $A$, then dom $R = $ range $R = A$
If $ R $ is belonging, between $A$ and $ $$\mathcal{P}$$ (A) $, then dom $R = A$ and ran $R = $$\mathcal{P}$$ (A) - \{\varnothing\}$
If $R$ is a relation included in a Cartesian product $A\times B$ (so that dom $R \subset A$ and ran $R \subset B$), it is sometimes convenient to say that $R$ is a relation from $A$ to $B$; instead of a relation from $A$ to $A$ we may speak of a relation in $A$. A relation $R$ in $A$ is reflexive if $x R x$ for every $x$ in $A$; it is symmetric if $x R y$ implies that $y R x$; and it is transitive if $x R y$ and $y R z$ imply that $x R z$.
FUNCTIONS
If $A$ and $B$ are sets, a, function from (or on) $A$ to (or into) $B$ is a relation $f$ such that dom $f = A$ and such that for each $x$ in $A$ there is a unique element $y$ in $B$ with $(x, y) \in f $. The uniqueness condition can be formulated explicitly as follows: if $(x, y) \in f $ and $(x, z) \in f $, then $y = z$. For each $x$ in $A$, the unique $y$ in $B$ such that $(x, y) \in f $ is denoted by $f(x)$. From now on, if $f$ is a function, we shall write $f(x) = y$ instead of $(x, y) \in f $ or $x f y$. The element $y$ is called the value that the function $f$ assumes (or takes on) at the argument $x$; equivalently we may say that $f$ sends or maps or transforms $x$ onto $y$. The words map or mapping, transformation, correspondence, and operator are among some of the many that are sometimes used as synonyms for function. The symbol $f:A \rightarrow B $ is sometimes used as an abbreviation for "$f$ is a function from $A$ to $B$." The set of all functions from $A$ to $B$ is a subset of the power set $ $$\mathcal{P}$$(A \times B)$; it will be denoted by $B^{A}$.
The domain of a function $f$ from $A$ into $B$ is, by definition, equal to $A$, but its range need not be equal to $B$; the range consists of those elements $y$ of $B$ for which there exists an $x$ in $A$ such that $f(x) = y$. If the range of $f$ is equal to $B$, we say that $f$ maps $A$ onto $B$. If $E$ is a subset of $A$, we may want to consider the set of all those elements $y$ of $B$ for which there exists an $x$ in the subset $E$ such that $f(x) = y$. This subset of $B$ is called the image of $E$ under $f$ and is frequently denoted by $f(E)$. (this means we consider the set of values of $f$ at the elements of $E$ ). Note that the image of $A$ itself is the range of $f$; the "onto" character of $f$ can be expressed by writing $f(A) = B$.
If $A$ is a subset of a set $B$, the function $f$ defined by $f(x) = x$ for each $x$ in $A$ is called the inclusion map (or the embedding, or the injection) of $A$ into $B$. The inclusion map of $A$ into $A$ is called the identity map on $A$.
If $f$ is a function from $B$ to $C$, say, and if $A$ is a subset of $B$, then there is a natural way of constructing a function $g$ from $A$ to $C$; define $g(x)$ to be equal to $f(x)$ for each $x$ in $A$. The function $g$ is called the restriction of $f$ to $A$, and $f$ is called an extension of $g$ to $B$; The definition of restriction can be expressed by writing $(f \mid A)(x) = f(x)$ for each $x$ in $A$; observe also that ran $(f \mid A) = f(x) $.
A function that always maps distinct elements onto distinct elements is called one-to-one (usually a one-to-one correspondence).
INVERSE FUNCTION
Associated with every function $f$, from $A$ to $B$, say, there is a function from $ $$\mathcal{P}$$ (A) $ to $ $$\mathcal{P}$$ (B)$, namely the function (frequently called $f$ also) that assigns to each subset $E$ of $A$ the image subset $f(E)$ of $B$.
Given a function $f$ from $A$ to $B$, let $f^{-1}$, the inverse of $f$, be the function from $ $$\mathcal{P}$$ (B)$ to $ $$\mathcal{P}$$ (A) $ such that if $D \subset B$ , then $f^{-1}(D) = \{x \in A \mid f(x) \in D \}$.
In words: $f^{-1}(D)$ consists of exactly those elements of $A$ that $f$ maps into $D$; the set $f^{-1}(D)$ is called the inverse image of $D$ under $f$. A necessary and sufficient condition that $f$ map $A$ onto $B$ is that the inverse image under $f$ of each non-empty subset of $B$ be a non-empty subset of $A$.
A necessary and sufficient condition that $f$ be one-to-one is that the inverse image under $f$ of each singleton in the range of $f$ be a singleton in $A$, alternatively the function whose domain is the range of $f$, and whose value for each $y$ in the range of $f$ is the unique $x$ in $A$ for which $f(x) = y$. In other words, for one-to-one functions $f$ we may write $f^{-1}(y) = x$ if and only if $f(x) = y$. Note that if $D \subset B$ then $f(f^{-1}(D)) \subset D$
COMPOSITE FUNCTION
If $f$ is a function from $A$ to $B$ and $g$ is a function from $B$ to $C$, then every element in the range of $f$ belongs to the domain of $g$, and, consequently, $g(f(x))$ makes sense for each $x$ in $A$. The function $h$ from $A$ to $C$, defined by $h(x) = g(f(x))$ is called the composite of the functions $f$ and $g$; it is denoted by $ g \circ f $ or, more simply, by $gf$,
Examples of Functions
- $f :$$\mathcal{R}$$ \rightarrow $$\mathcal{R}$$ $ defined by polynomials, such as the example $f (x) = x^{2} + x + 1$
- The rational functions $f :$$\mathcal{R}$$ \rightarrow $$\mathcal{R}$$ $ such as $f(x) = \dfrac{x^{3}-4x^{2}+x -11}{(x^{2}+1)^{5}}$
- The absolute value $f :$$\mathcal{R}$$ \rightarrow $$\mathcal{R}$$+ $ such as $f(x) = \cases{x & \text{ if } x \geq 0 \\ -x & \text{ if } x < 0}$
- Analytic functions from $f :$$\mathcal{R}$$ \rightarrow $$\mathcal{R}$$ $ such as $f(x) = \sin{x}, f(x) = \log{x}$
Sample Quiz Questions
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