Relations in DMS & types of discrete mathematical structures.

1. Relations -Discrete Mathematics
Ms.Rachana Pathak
(rachanarpathak@gmail.com)
Assistant Professor, Dept of Computer Science and Engineering
Walchand Institute of Technology, Solapur
(www.witsolapur.org)
TARGET GATE CSE / IT 2019

2. Learning Outcome
2Walchand Institute of Technology, Solapur
At the end of this session,
Students will be able to explain Relations – domain and range, and solve
problems based on Relations.

3. Prerequisite
• Basics of Discrete Mathematics
• Students from CSE/IT Engineering branch
• Students appearing for GATE (CS)
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4. Introduction
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What is a
relation?

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Father to a Son
Father to
a
daughter
Mother
to a Son
Mother to a Daughter

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Let us describe in terms of Discrete Mathematics.

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8. So, this can be further ordered as -
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This is said ordered pair as you cannot
reverse or change the order.
Eg. Son cannot be inversely related as
father of father.

9. Think & Write
Question : Can we show relation between son & daughter?
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10. Answer
No, because they are elements of the same set.
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11. Definition
• Any set of ordered pairs defines a binary relation or simply a
relation. Binary relations represent relationships between
elements of two sets.
• If R is a relation, a particular ordered pair, (x,y)∈ R can be
written as xRy and can be read as “x is in relation to R with y”
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12. Examples
• The relation “greater than” for real numbers is denoted by ‘>’.
• If x and y are any two real numbers such that x>y,
then we say that (x,y) ∈ >.
Thus, the relation > is
> = {(x,y) | x and y are real numbers and x>y}
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This method of
representation is
called as set
builder method
‘|’ symbol is
used to
represent “such
that”

13. Domain and Range of a Relation
• Domain: Set of all the first numbers of the ordered pairs.
In other words, the domain is all of the x values.
• Range: Set of the second numbers in each pair, or the y-values
• Example : Domain and Range of a RELATION
{(0,1) , (3,22) , (90, 34)}
Domain : 0, 3, 90
Range: 1, 22 ,34
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14. Examples
• List the domain and the range of the relation
• {(-3,3) ; (0, 2) ; (3, 3) ; (6, 4) ; ( 7, 7)}
• Domain: {-3, 0, 3, 6, 7}
• Range: {3, 2, 4, 7}
Representation : For a Set S
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Even though the number
3 is listed twice in the
relation, you only note
the number once when
you list the domain or
range
D(S) = {x | (∃y) ((x,y) ∈ S}
R(S) = {y | (∃x) ((x,y) ∈ S}

15. Conclusion :
In this session, We have studied all about Relations.
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16. References
• 1. Discrete mathematical structures with applications to computer science -- J. P.
Tremblay & R. Manohar (MGH International)
• Reference Books:
• 1. Discrete Mathematics with combinatorics and graph theory- S. SNTHA
(CENGAGE Learning)
• 2. Discrete Mathematical Structures – Bernard Kolman, Robert C. Busby (Pearson
Education)
• 3. Discrete mathematics -- Liu (MGH)
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17. Thank You !!!
Walchand Institute of Technology, Solapur