Question

To determine whether the relationon the set of all people is reflexive, symmetric, anti symmetric, transitive, where $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$ if and only if aand $\mathit{b}$have a common grandparent.

Short answer

The given set is reflexive and symmetric.

Step-by-step solution

Given data

The given set is a number of people $a,b$.

Concept used of relation

A relation $\mathbf{\text{R}}$on a set $\mathbf{\text{A}}$is called reflexive if $\mathbf{(}\mathit{a}\mathbf{,}\mathit{a}\mathbf{)}\mathbf{\in }\mathit{R}$for every element $\mathit{a}\mathbf{\in }\mathit{A}$.

A relation $\mathbf{\text{R}}$on a set $\mathbf{\text{A}}$is called symmetric if $\mathbf{(}\mathit{b}\mathbf{,}\mathit{a}\mathbf{)}\mathbf{\in }\mathit{R}$whenever $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$, for all $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{A}$

A relation $\mathbf{\text{R}}$on a set $\mathbf{\text{A}}$such that for all $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{A}$, if $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$and $\mathbf{(}\mathit{b}\mathbf{,}\mathit{a}\mathbf{)}\mathbf{\in }\mathit{R}$then $\mathit{a}\mathbf{=}\mathit{b}$is called anti symmetric.

A relation $\mathbf{\text{R}}$on a set $\mathbf{\text{A}}$is called transitive if whenever $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$and $\mathbf{(}\mathit{b}\mathbf{,}\mathit{c}\mathbf{)}\mathbf{\in }\mathit{R}$then $\mathbf{(}\mathit{a}\mathbf{,}\mathit{c}\mathbf{)}\mathbf{\in }\mathit{R}$for all $\mathit{a}\mathbf{,}\mathit{b}\mathbf{,}\mathit{c}\mathbf{\in }\mathit{A}$

Solve for relation

A has a common grandparent with itself. If a has a common grandparent with b, then clearly b has a common grandparent with a. It is not transitive: suppose a has a common grandparent with b, and b has a common grandparent with c. a's common grandparent with b might be b's grandparent on b's mother's side, while b's common grandparent with c might be on b's father's side; thus a and c need not necessarily have a common grandparent. It is not anti symmetric: a and b need not be the same person if a has a common grandparent with b and b a common grandparent with a.

The given set is reflexive and symmetric.