Is this relation transitive? $R=\{(1,2),(1,1),(2,1),(2,2)\}$ over $A=\{1,2,3\}$

Question

Is this relation $R$ over $A$ transitive?$$A=\{1,2,3\}$$ $$R=\{(1,2),(1,1),(2,1),(2,2)\}$$

Since from the definition a relation is transitive if $\forall x,y,z\in A (xRy,yRz\to xRz)$, so since $3$ isn't in the relation then it isn't transitive?

Answer 1

This relation is transitive. 3 doesn't play any role here since you don't require a transitive relation to be full. Observe that the transitivity here means: $$(1,2)\wedge(2,1)\in R\Rightarrow(1,1)\in R \\(2,1)\wedge(1,2)\in R\Rightarrow(2,2)\in R$$ and these are found in the relation so it's transitive. In fact that's equivalence relation on $\{1,2\}$ (since it's also reflexive and symmetric there).

EDIT: We can justify my remark by definitions ($T$ is the set on which the relation $R$ is defined):

• Reflexivity means $\forall x\in T,(x,x)\in R$ - In our case $(1,1),(2,2)\in R$
• Symmetry means $xRy\Leftrightarrow yRx$. It doesn't mean that the relation is full. In our case $(1,2)\in R\wedge (2,1)\in R$. Trivially $(1,1),(2,2)\in R$.

So for these reasons it's an eqivalence relation on $T=\{1,2\}$.