Answer to Question #117120 in Discrete Mathematics for Priya

Question

Consider the two relationsdefined on the set of all people. (i) (a,b)  R, iff a is taller than b (ii) (a,b)  R, iff a and b were born on the same day. Determine whether the relations are reflexive, symmetric, antisymmetric and/or transitive

Answer 1

#Q117120

1) (a,b) i.e. aRb

Let h1 and h2 height of two people a and b respectively

and aRb this implies h1>h2

a) Reflexive i.e. aRa

But h1> h1 ( a contradiction)

Hence relation is not reflexive

b) Symmetric i.e. aRb implies bRa

Hence if h1>h2 doesn't imply h2>h1

For example : let h1=7 and h2=3

Here 7>3 doesn' t imply 3>7

Hence relation is Not symmetric

c) Antisymmetric : if aRb imply bRa then a=b

h1>h2 and if h2> h1 then it does't imply h1=h2

Hence Not anti-symmetric

d) Transitivity : aRb and bRc imply aRc

Let h1,h2,h3 height of three people if h1> h2 and h2> h3 it imply that h1>h3

For exp : 9>6 ,6>5 imply 9>5

Hence transitivity hold

Ans. So, in (1) part only Transitivity hold

2) (a,b) i.e. aRb iff a and b born on the same day

a) Reflexive aRa

Clearly one person born on the same day

Hence R is reflexive.

b) symmetric i.e. aRb imply bRa

aRb i.e. a and b born on the same day imply b and a born on the same day

i.e. aRb imply bRa

Hence R is symmetric

c) Anti-symmetric

aRb and bRa doesn't imply a=b

For exp : there is two person whose birth day is same

Hence Not anti-symmetric

d)Transitivity : aRb ,bRc then aRc

If birth day of a and b are same , b and c have same birth day imply a and c also have same birth day

For exp: let a and b have birth day suppose 2 june

then c birth day is also 2 june because b and c has same birth day

Hence a and c also same birth day

Hence transitivity hold

Ans. Hence 2) relation is reflexive , symmetric ,transitivity but NOT anti-symmetric