Jain College of Engineering, Belagavi

Department of Mathematics
Module 3: Relations and Functions
Assignment – 1
Course Name: Discrete Mathematical Structures Course Code: BCS405A
Semester : IV Maximum Marks : 20
Date: 03/03/2025 Date of Submission:
Question C P R
Q.NO B
O O
T
Define Relations and Functions. Let function �: � → � be defined by � � = �2 + 1 find the
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images of the subsets i) � = 2, 3 ii) −2, 0, 3 iii) 0, 1 iv) −6, 3
3� − 5 ��� � > 0
Let function �: � → � be defined by � � = then find
−3� + 1 ��� � ≤ 0
2 i) � 0 , � −1 , �
5 5
, �( − 3 ) ii) �−1 0 , �−1 1 , �−1 −1 , �−1 3 , �−1 −3 , �−1 ( − 6) 3 1 L1
3
iii) �−1 −5, 5 , � ( −1 −6, 5
)
The functions �: � → � and �: � → � are defined by � � = 3� + 7 ��� ��� � ∈ � , and g � =
3
� �3 − 1 ��� ��� � ∈ �. Verify that � is one-to-one but g is not.
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The function �: � � � → � is defined by � �, � = 2� + 3�. Verify that � is on-to but one-to-one.
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State pigeonhole principle and its generalization form. Prove that any set of 29 persons at least five
5
persons must have been born on the same day of the week.
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6 Show that if n+1 numbers are chosen from 1 to 2n then at least one pair add to 2n+1. 3 1 L1
7 If f:A→B, g:B→C, h:C→D:are three functions, then Prove that ho(gof)=(hog)of. 3 1 L1
Consider the function � and � defined by � � = �3 and � � = �2 + 1 ��� ��� � ∈ �. Find
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���, ���, �2 , �2 .
Let f and g be functions from R to R defined by f(x)=ax+b and g � = 1 − � + �2 , If ��� � =
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9�2 − 9� + 3 determine a and b
0, if x is even
Let f,g,h be functions from Z to Z defined by f(x)=x-1, g(x)=3x, h(x) =
10 1, if x is odd
Determine (fo(goh))(x),((fog)oh)(x) and verfify that fo(goh)=(fog)oh
Let � = {1,2,3,4,6} and � be the relation on � defined by ��� if and only if “� is multiple of �",
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Represent R as a set of ordered pairs and matrix , draw its digraph.
Define equivalence relation. Let � = {1,2,3,4,5,6,7,8,9,10,11,12} on this set the relation R defined
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by �, � ∈ � if and only if “� − � is multiple of 5". Verify R is an equivalence relation.
For a fixed integer � > 1, prove that relation “ congruent modulo n” is an equivalence relation on
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the set of all integers Z.
Let � = {1,2,3,4,6,12,18} let � be the relation on � defined by ��� if and only if “� divides �",
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written by �|�. Draw Hasse diagram of the Poset (A, R). Determine the relation matrix for R.
15 Draw the Hasse (POSET) diagram which represents positive divisors of 36. 3 1 L1
Consider the Hasse diagram for the poset (A, R) given below. If � = {�, �, �} then find
i) i) all upper bounds of B iii) The least upper bounds of B
ii) ii) All lower bounds of B iv) The greatest lower bound of B
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iii)
*CO- Course Outcome Apply basic concepts of relations, functions and partially ordered sets for computer representation

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