2.5. Types of Relations

A binary relation ^ defined on an unempty set ^ is called reftexive if ^^^ for every ^ E ^, that is, if ^) E ^ for every ^ E ^.

Ex^wp/e 2.J.7. Given the following five relations.

1. Relation < (is less than or equal to) on the set Z of integers;

2. Set inclusion c on a collection C of sets;

3. Relation ± (perpendicular) on the set L of lines in a plane;

4. Relation ( parallel) on the set L of lines in a plane;

5. Relation of divisibility on the set A^ of positive integers. (Recall x y if there exists z such that xz = y.)

Determine which of these relations are reflexive.

A binary relation ^ on a set ^ is called irreftexive if ^)% ^ for

all ^ E ^.

A binary relation ^ on a set ^ is called symmetric if whenever ^^^ then that is whenever ^) E ^ then ^) E ^.

A binary relation ^ on a set ^ is called antisymmetric if whenever ^^^ and ^^^ then ^ = that is, if ^ # ^ and ^^^ then

De//'м/Y/oм. A binary relation ^ on a set ^ is called transitve if whenever ^^^ and then ^^c, that is, if whenever ^), c) E ^ then c) E

De//'м/Y/oм. A binary relation ^ on a set ^ is called complete if whenever ^ E ^ and ^ E R then ^ = or ^) E or ^) E

E.^wp/e 2.J.2. Consider the following five relations on the set ^ = {1,2,3}: ^ = {(1,1), (1,2(1,3), (3,3)}, ^ = {(1,1), (1,2(2,1), (2,2), (3,3)}

r = {(1,1), (1,2), (2,2), (2,3)}, 0 - empty relation, ^ x ^ - universal relation.

Determine whether or not each of the above relations on ^ is:

1) reflexive; 2) symmetric; 3) transitive; 4) antysymmetric.

^o/MY/oM

1. ^ is not reflexive since 2 E ^ but (2,2)% r is not reflexive since (3,3)% r and,

similarly, 0 is not reflexive. ^ and ^ x ^ are reflexive.

2. ^ is not symmetric since (1,2)E ^ but (2,1)% ^, and similarly r is not symmetric.

0, and ^ x ^ are symmetric.

3. r is not transitive since (1,2) and (2,3) belong to but (1,3) does not belong to r. The other four relations are transitive.

4. ^ is not antisymmetric since 1 ^ 2 and (1,2) and (2,1) both belong to Similarly, ^ x ^ is not antisymmetric. The other three relations are antisymmetric.

2.6. Functional Relations

A section .x = a of a set ^ is a set of elements y e R for which (a, y)e This section is denoted by ^(a).

Let c = (a, ^), where c e ^ x R. An element a is called a projection of an element c on a set and denoted by Pr^ c = a.

Exaw^/e 2.d.7. Let ^ = {a₁, a ₂, a ₃, a ₄, a₅}, R = , ^3, } and the relation

^ = ^{(a1^, ^2 ^{), (a}1^, ^4 ^{), (a}2^{, ), (a}2^{, ), (a} 3^, ^2 ^{), (a}3^, ^3 ^{), (a}3^, ^4 ^{), (a} 5^{, ), (a}5^, ^3 ^{)} be}

given. Find:

1. sections .x = a^ = 1,5 );

2. Pr^ (a₂, ) and Pr^^ .

Using the definitions of a section and a projection, we have

1. ^(a1 )={^2, ^4}; ^(a2 )={^1, ^3}; ^(a3 )={^2, ^3, ^4}; ^(a4 )=0;

^(a5 )={^1, ^3}.

2. ^Pr.4 ^(a 2^, ^3 ⁾= ^a 2; ^Pr.4 ^ = ^{a1^{, a}2^{, a}3^{, a}5 ^}.

De/'M'^'oM. A relation ^ c ^ x R is called a functional relation if for each xx e ^ a section ^ with respect to xY contains not more than one element y e R or none. Such relation is called a function from ^ into R and denoted by /: ^ ^ R which is read: " / is a function ^om ^ into R ".

De/'M'^'oM. If the function/ is defined on a set D c ^ then this set D is called the domain of definition of or more briefly the domain of /.A subset Im c R, where Im = {/ (x) x e D} is called the range or image of

De/'M'^'oM. An element ^ = /(a), where a e D is called an image of the element a, and element a is called a prototype of the element

De/'M'^'oM. If D = ^, then a function/ is called everywhere defined. Frequently a function can be expressed by means of mathematical formula. For example, consider the function which sends each real number into its square. We can describe this function by writing

/ (x )= x ² or y = x ²

In the first notation, xx is called a variable and the letter / denotes the function. In the second notation, xx is called the independent variable and y is called the dependent variable since the value of y will depend on the value of xx.

Every function / : ^ ^ R gives rise to a relation from ^ to R called the graph of y and denoted by

Graph of/= {(a, ^) a e ^ = /(a)}.

2.7. One-to-one, onto, and Invertible Functions

A function / : ^ ^ R is said to be one-to-one if different elements in the domain ^ have distinct images.

A function / : ^ ^ R is said to be an onto function if each element of R is the image of some element of

In other words, / : ^ ^ R in onto if the image of / is the entire range, i.e. if /(^)= R. In such a case we say that /is a function ^om ^ onto R or that f maps ^ onto R.

A function / : ^ ^ R is invertible if its inverse relation /"¹is a function from R to

In general, the inverse relation / may not be function. In what follows we use the terms injective for one-to-one function, surjective for an onto function, and bijective for a one-to-one correspondence.