Let T be a relation from A = {0, 1, 2, 3} to B = {0, 1, 2, 3, 4} such that (a, b) T iff b2 – a2 is an odd number. (A, B U = Z.) (Hint: Write down all the elements of T. For example, if 4 B and 1 A then 42 – 12 = 16 – 1 = 15 which is an odd number, thus (1, 4) T.) Answer questions 5 and 6 by using the defined relation T. Question 5 Which one of the following alternatives provides only elements belonging to T? 1. (3, 1), (4, 1), (3, 2) 2. (0, 1), (2, 4), (2, 3) 3. (3, 0), (1, 2), (3, 4) 4. (1, 0), (1, 2), (1, 3) Question 6 Which one of the following statements regarding the relation T is true? 1. T is transitive. 2. T is symmetric. 3. T is antisymmetric. 4. T is irreflexive.
let T be a relation from A ={0,1,2,3} to B={0,1,2,3,4}
such that
(a,b)∈T iff b2-a2 is an odd number.
(A,B)⊆U=Z)
Question5
which one of the following alternatives provides only elements belonging to T?
1. (3,1),(4,1),(3,2)
2. (0,1),(2,4),(2,3)
3. (3,0),(1,2),(3,4)
4. (1,0),(1,2),(1,3)
Question6
which one of the following statements regarding the relation T is true?
1. T is transitive
2. T is symmetric
3. T is antisymmetric
4. T is irreflexive
solution:-
firs of all we write all (a,b)=b2-a2 ,and select element of T,which b2-a2=odd.
(0,0)=0 (0,1)=1 (0,2)=4 (0,3)=9 (0,4)=16
(1,0)=-1 (1,1)=0 (1,2)=3 (1,3)=8 (1,4)=15
(2,0)=-4 (2,1)=-3 (2,2)=0 (2,3)=5 (2,4)=12
(3,0)=-9 (3,1)=-8 (3,2)=-5 (3,3)=0 (3,4)=7
we create a set of element which show b2-a2=odd.
T={(0,1) (0,3),(1,0),(1,2),(1,4),(2,1),(2,3),(3,0),(3,2),(3,4)}
answer5
elements of option 3 contains in the T set.
so option 3 is correct answer.
answer6
1. T is transitive:-
definition of transitive:-
if (a,b) where b2-a2=odd and (b,c) where c2-b2=odd
then (a,c) where c2-a2=odd.
it property called transitive.
check:-for (a,c)
c2-a2=(c2-b2)+(b2-a2)
=odd+odd
=even {we know that sum of two odd is even always}
so T is not transitive
2. T is symmetric
definition of symmetric
if (a,b) where b2-a2=odd
then (b,a) where a2-b2=odd.
it property called symmetric
check:-for (b,a)
a2-b2=-(b2-a2)
=-(odd)
=odd {negative odd is also odd}
so T is symmetric
3. T is antisymmetric
option 2 is correct hence option 3 is incorrect.
4. T is irreflexive
definition of reflexive
if (a,a) where a2-a2=odd
it property called reflexive
check:-for (a,a)
a2-a2=0 {0 is not odd}
so T is not reflexive
hence T is irreflexive
here option 2 and 4 are correct option
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