prove why each relation has or does not have the properties

prove why each relation has or does not have the
properties: reflexive, symmetric, anti-symmetric, transitive

Let A = {set of all people}, relation R: A times A where R = {(a, b) | a is at least as tall as b} Set S = {0, 1, 2, 3}, relation R: S times S is defined as: (m, n) element R if m + n = 4;

Solution

1.

aRa because a is at least as tall as himself. So R is reflexive

R is not symmetric because a could be stricly taller than b then aRb but bRa is not true

Let, aRb and bRc

So, a is at least as tall as b, b is at least as tall as c

HEnce, a is at least as tall as c

Hence, aRc

So, R is transitive

It is not antisymmetric because there can be two different people of same height,say a and b

Then, aRb and bRa are true

2.

R is not reflexive as    (3,3) does not belong to R

R is symmetric because m+n =4 implies n+m=4

R is not transitive

(1,3) and (3,1) are in R

but (1,1) is not