The document provides a comprehensive overview of relations in mathematics, including their definition, types, and properties. It covers various types of relations such as reflexive, identity, symmetric, empty, universal, transitive, and inverse relations, along with examples of each. The presentation also outlines the contributions of different group members in preparing for the presentation.

1. RELATION
Presentation of Group 1

2. LET’S PLAY A
GAME
FIND ME🔍

4. •To comprehend the concept of
relations, their types, and how they
represent connections between
elements.
•To explore and analyze the properties
of relations
OBJECTIVES

5. RELATION
A relation is a set of ordered pairs,
where each ordered pair consists of
elements from one set related to
elements from another set. These
ordered pairs represent connections
or relationships between the
elements.
• It's a rule that pairs
each element in one
set, called the
domain. with one or
more elements from
a second set called
the range . It creates
a set of ordered pairs.

6. E.g.
R= {(1,2), (2,4), (3,6), (4,8),
(5,10)}
1,2,3,4,5-domain
2,4,6,8,10- called
range

7. let A={1,2} and ={1,2,3} and define a relation R from A and B as given that any (x,y) AxB,
∈
(x,y) R means that x-y/2 is an integer.
∈
1.State explicity which ordered pairs are in AxB and which are in R
AxB= {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)}
(1,1) R because 1-1/2 is equal to 0/2= 0
∈
(1,2) R because 1-1/2 is equal to -1/2
∉
(1,3) R because 1-3/2 is equal to -2/2 = -1
∈
(2,1) R because 2-1/2 is equals to 1/2
∉
(2,2) R because 2-2/2 is equals 0/2 = 0
∈
(2,3) R because 2-3/2 is equals to -1/2
∉
DOMAIN- (1,2)
RANGE- (1,2,3)

9. TYPES
OF
RELATION
•Reflexive relation
•Identity relation
•Symmetric relation
•Empty relation
•Universal relation
•Transitive relation
•Inverse relation

10. REFLEXIVE RELATION
•A reflexive relation is a binary relation on
a set for which every element is related to
itself.
•Relation (R) on a set A is called reflexive.
For example, consider a set A= {1,2}.
Now, the reflexive relation will be R=
{(1,1), (2,2), (1,2), (2,1)}. Thus, a relation is
reflexive if: (a,a) R a A
∈ ∀ ∈

11. IDENTITY RELATION
• In identity relation, every element of set A is related to itself
only. I = {(a, a), A}.
∈
• For instance, if we have a set A = {1, 2, 3}, the identity
relation on set A would be {(1, 1), (2, 2), (3, 3)}, where each
element is related to itself.

12. SYMMETRIC
RELATION
• Any relation R on a set A is said to be symmetric.
Hence, whenever (a, b) € R (b, a) € R
⟹ , it is a
symmetric relation.
• For example, a set A= {1, 2, 3}. Therefore, R= {(1, 1),
(1, 2), (1, 3), (2, 3), (3, 1), ( 2, 1), (3, 2)}

13. EMPTY RELATION
• An empty relation (or void relation) is one in
which there is no relation between any
elements of a set. For empty relation, R = Ø ⊂
A x A.
• For example, if set A = {1, 2, 3} then, one of the
void relations can be R = {(a, b): a + b= 8}.
Therefore, R= Ø or null set.

14. UNIVERSAL RELATION
• A universal (or full relation) is a type of relation in which
every element of A set is related to each other. Consider set.
For universal relation, R = A × A.
• In other words, a relation is universal if every set is subset
of itself.
• If we have a set A = {1, 2}, then, the A x A= {(1, 1), (1, 2), (2,
1), (2, 2)}. Therefore, R= A x A

15. TRANSITIVE RELATION
• If element A is related to element B, and element B is
related to element C, then element A is also related to
element C. An (a, b) R and
∈ (b, c) R (a, c) R,
∈ ⟹ ∈ ∀
a, b, c A.
∈
• In other words, if (a, b) and (b, c) are in the relation,
then (a, c) must also be in the relation.
• For example, consider the relation "is taller than"
among a group of people. If person A is taller than
person B, and person B is taller than person C, then it
follows that person A is also taller than person C.

16. INVERSE RELATION
• The inverse of a relation is a relation obtained by
interchanging or swapping the elements or coordinates of
each ordered pair in the relation. Inverse relation in sets can
be defined using the ordered pairs. The domain and range
of an inverse relation can be written by swapping the
domain and range of that relation. That means the domain
of relation will be the range of its inverse, and the range of
relation will be the domain of its inverse.
• For example, R= {(0, 2), (5, 7), (6, 9)}
R= {(2,0), (7, 5), ( 9, 6)}

17. Group 1
*Dayto, Kristella- Give suggestion, informations, ppt, report
•Barnuevo, Cielo- give information, ppt
•Edma, Christine- give information, ppt
•Coronel, Geoffrey- contributed in prize , host in the game
•Borja, Alan- contributed in prize ,host in the game
•Cano, Angela - manila paper, report
•Cabradilla, Carmela- contributed in prize
•Alabat, Thea - give information, report
•Agotilla, Lourdes- suggested on what to put in the ppt, report
•Belgica, Kristel Joy- give information, ppt, and report
•Camano, Ailyn- give information
• Ancajas, Cristal - report
• Dela Cruz- report
• De Luna- report
• Arraz, Jerlyn Nikka- contributed in prize

18. THANK
YOU!