# reflexive ,symmetric , antisymmetric ,transitive

Let H be the half-sibling relation on the set of all people in the world. Is H reflexive? Is H symmetric? Is H antisymmetric? Is H transitive?

Can anyone answer the above questions with reasoning just to understand the way of thinking?

• Have you tried anything? These things follow from the definitions of these words. Do you know what each word means? Do you know what a half-sibling is? Nov 6, 2017 at 22:46

Reflexive -- can you be your own half sibling?
Symmetry -- If Tom is your half-brother are you Tom's half-sibling?

Transativity -- This is a little trickier. Suppose Tom is your half-brother -- you have the same father and different mothers, and Suzy is Tom's half-sister, Tom and Suzy have the same mother and different fathers, are you related to Suzy?

Anti-symmetric -- this is a more difficult concept. If a member is related to a different member $R(a,b), a\ne b$ then the symmetric $R(b,a)$ is never true. An example of an anti-symmetric relation would be parent-child.

• If Tom is your half-sibling are you Tom's half-sibling? ("Half-brother" is not the same.) Nov 6, 2017 at 22:51
• okay so: a)NOT reflexive because I cant be my own half-sibling. b)YES Symmetric because f H(x,y) then H(y,x). So if y is half sibling of x,x will be also hallf sibling of y. c)NOT Transitive Nov 6, 2017 at 23:04
• Reflexive means $H(x,x)$ is true. So is a person a half-sibling of himself?
• Symmetric means if $H(x,y)$ then $H(y,x)$. So if $x$ is a half-sibling of $y$, then is $y$ a half-sibling of $x$?

Can you add definitions of the other 2 and finish the problem?

• Transitive means: H(x,y) and H(y,z) --> H(x,z). Thus it is not transitive because regarding the example in the above comments I am not related with suzy. Antisymmetric: means H(x,y) ∈ R and H(y,x) ∈ R implies x = y. Thus it is not antisymmetric. Nov 6, 2017 at 23:18