Discrete Mathematics Assignment Solutions

DISCRETE MATHEMATICS
Assignment
[Pick the date]
Student Name

Question 1
(a) Smallest equivalence relation on the set { a , b , c , d ,e } that contains the relation
{ ( a ,b ) , ( a , c ) , ( d , c ) }
It can be said that an equivalence relation contains (a, b) then due to presence of symmetric it
will also contains (b, a).
Similarly,
R contains (a, a) (Transitive)
R contains (a, c) and also contains (c, a) (Symmetric)
R contains (c, a) and (a, b) contains (c, b) (Transitive)
R contains (b, c) (Symmetric)
R contains (d, e) and also contains (e, d) (Symmetric)
R contains (d, e) and also contains (e, d) (Transitive)
R contains (b, b) and (e, e), (c, c), (d, d) and (e, e)
Hence,
R= { ( a , a ) , ( b , b ) , ( c , c ) , ( d , d ) , ( e , e ) , ( a , b ) , ( b , a ) , ( a , c ) , ( c , a ) , ( c , b ) ( b , c ) , ( e , d ) , ( d , e ) }
(b) Equivalence classes of the equivalence relation
Equivalence classes are [a] and [d]
[ a ]= { a , b , c }= [ b ]=[c ]
[ d ] = { d , e }= [ d ] =[e]
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Question 2
Partial order p be defined on Set S= {1,2 , 3,4,5,6,7,8,9,10,11,12 }
(a) Hasse diagram for partial order p
(b) Largest set of mutually comparable elements in p
{1 , 2 , 4 , 12 }
(c) Largest set of mutually non-comparable elements in p
{2 , 3 , 5 ,7 , 11}
Question 3
Let S4 is group consisting the set of all bijections from { 1 , 2, 3,4 } ¿ {1,2,3,4 }
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