Relation Set

1. Relation

2. Representing sets using
computer
• 1 = true , 0 = false
• 10 1010 1010 = {1,3,5,7,9}
• Union = OR
• Intersection = AND
• Complement = inverse all

3. Concept of relation between two
sets
• If we want to describe a relationship between elements of
two sets A and B, we can use ordered pairs with their first
element taken from A and their second element taken from B.
• Since this is a relation between two sets, it is called a binary
relation.
• Definition: Let A and B be sets. A binary relation from A to B is
a subset of A B.
• In other words, for a binary relation R we have R A B.
• We use the notation aRb to denote that (a, b) R and aRb to
denote that (a, b) R.
• When (a, b) belongs to R, a is said to be related to b by R.

4. example
• Example: Let P be a set of people, C be a set of cars, and D be
the relation describing which person drives which car(s).
P = {Carl, Suzanne, Peter, Carla}
C = {Mercedes, BMW, tricycle}
D = {(Carl, Mercedes), (Suzanne, Mercedes),
(Suzanne, BMW), (Peter, tricycle)}
• This means that Carl drives a Mercedes, Suzanne drives a
Mercedes and a BMW, Peter drives a tricycle, and Carla does
not drive any of these vehicles.

5. Properties of relations
• Reflexive
• A relation R on a set A is called reflexive if (a, a) R for every
element a A.
• Example:
• R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4),
(4, 4)}
• Answer: R is reflexive because they both contain all pairs of
the form (a, a),namely, (1, 1), (2, 2), (3, 3), and (4, 4).

6. • Irreflexive
• A relation on a set A is called irreflexive if (a, a) R for every
element a A.
• Example:
These 4 irreflexive relations are :
1. Empty
2. {(1,2)}
3. {(2,1)}
4. {(1,2), (2,1)}

7. • Symmetric
• A relation R on a set A is called symmetric if (b, a) R
whenever (a, b) R for all a, b A.
• Example:
• R={(1, 1), (1, 2), (2, 1)}
• Answer: R is symmetric.Both (2, 1) and (1, 2) are in the
relation so R is symmetric.

8. • Antisymmetric
• A relation R on a set A is called antisymmetric if a = b
whenever (a, b) R and (b, a) R.
• Example:
• R={(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}
• Answer:R is antisymmetric.There is no pair of elements a and
b with a ≠ b such that both (a, b) and (b, a) belong to the
relation.

9. • Transitive
• A relation R on a set A is called transitive if whenever (a, b) R
and (b, c) R, then (a, c) R for a, b, c A.
• Example: Are the following relations on {1, 2, 3, 4} transitive?
• Answer: R = {(1, 1), (1, 2), (2, 2), (2, 1), (3, 3)}

10. Definition or concepts of function
on general sets
•In discrete mathematics ,functions are used
•in the definition of such discrete structures as
sequences and strings.
•used to represent how long it takes a computer to
solve problems of a given size.
Concept of function:
•Let A and B be nonempty sets.
•A function f from A to B is an assignment of exactly
one element of B to each element of A.
•We write f (a) = b if b is the unique element of B
assigned by the function f to the element a of A.
•If f is a function from A to B, we write f : A → B

11. •If f is a function from A to B, we say that A is the domain of f and B is
the codomain of f.
•If f (a) = b, we say that b is the image of a and a is a preimage of b.
•The range, or image, of f is the set of all images of elements of A.,
• if f is a function from A to B, we say that f maps A to B.

12. Concept of Boolean Function
• Definition 1 : A literal is a Boolean variable or its
complement. A minterm of Boolean variables x1, x2, . . . ,
xn is Boolean product y1 y2. . .yn where yi = xi or
Example1:
Find the minterm F such that F = 1 if x1 = x3 = 0 and F = 0 if
x2 = x4 = x5 = 1.
Solution: The minterm is
The sum of minterms that represents the functions is
called the sum-of-products expansion or the disjunctive
normal form of the Boolean function.

13. Example2: Find the sum-of-product expansion for the
function .
(1) Solution:

15. Injective functions
Injective means that every member of “A” has its own unique
matching member in “B”.
A function f is injective if and only if whenever f(x) = f(y), x =y.
It’s also called “one to one”.
Ex: {a, b, c, d} to {1,2,3,4,5} with f(a) =2, f(b) =1, f(c)=3, f(d)=4 is
one to one.
a 1
b 2
c 3
d 4
5

16. Surjective functions (an onto function)
Surjective means that every “B” has at least one matching “A” (maybe
more than one).
A function f (from set A to B) is surjective if and only for every y in B,
there at least one x in A such that f(x) = y, in other words f is
surjective if and only if f(A) = B.
Ex: f be the function from {a,b,c,d,e} to {1,2,3,4} defined by f(a)=2,
f(b)=1, f(c)=3, f(d)=3, f(e)=4. Is f is onto function?
Answer: NO
a 1
b 2
c 3
d 4
e

17. Bijective functions
Bijective means both Injective and Surjective together.
Perfect “one-to-one correspondence” between the members of the
sets is existed.
A function f (from set A to B) is bijective if, for every y in B, there is
exactly one x in A such that f(x) = y.
Ex: f be the function from {a,b,c,d,e} to {1,2,3,4,5} with f(a)=2, f(b)=1,
f (c)=6, f(d)=3, f(e)=5. Is f a bijection?
Answer: YES.
a 1
b 2
c 3
d 4
e 5

18. Definition and example of inverse
function
An inverse function, which we call f-1 is another function
that take y back to x. f(x)= y. So, f-1(y)= x.

19. • Example:
Let f: Z Z be such that f(x)=x+1
f(x)= x+1
y=x+1
y-1=x
f-1(y)= y-1

20. Definiton and example of
composition function
• Let g be a function from the set A to the set B and let f be a function from
the set B to the set C. The composition of the functions f and g, denoted
by f ◦ g, is defined by
(f ◦ g)(a) = f (g(a)).

21. • Example:
f (x) = 2x + 3 and g(x) = 3x + 2
• Solution:
(f ◦ g)(x) = f (g(x))
f (3x + 2) = 2(3x + 2) + 3 = 6x + 7
and
(g ◦ f )(x) = g(f (x))
g(2x + 3) = 3(2x + 3) + 2 = 6x + 11.