Relation Properties 89A280

1. **State the problem:** We have a set $A = \{a, b, c, d\}$ and a relation $R$ represented by the matrix $$\begin{bmatrix}1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1\end{bmatrix}$$ We need to determine which properties the relation $R$ satisfies: reflexive, anti-symmetric, transitive, or all. 2. **Recall definitions:** - Reflexive: For all $x \in A$, $(x,x) \in R$. In matrix terms, all diagonal entries are $1$. - Anti-symmetric: For all $x,y \in A$, if $(x,y) \in R$ and $(y,x) \in R$, then $x = y$. - Transitive: For all $x,y,z \in A$, if $(x,y) \in R$ and $(y,z) \in R$, then $(x,z) \in R$. 3. **Check reflexivity:** The diagonal entries are all $1$: $$r_{11} = 1, r_{22} = 1, r_{33} = 1, r_{44} = 1$$ So, $R$ is reflexive. 4. **Check anti-symmetry:** Look for pairs $(x,y)$ and $(y,x)$ with $x \neq y$ where both are $1$. - For example, $r_{12} = 1$ but $r_{21} = 0$. - $r_{13} = 1$, $r_{31} = 0$. - $r_{14} = 1$, $r_{41} = 0$. - $r_{23} = 1$, $r_{32} = 0$. - $r_{24} = 1$, $r_{42} = 0$. - $r_{34} = 1$, $r_{43} = 0$. No pair $(x,y)$ and $(y,x)$ with $x \neq y$ both equal to $1$. Thus, $R$ is anti-symmetric. 5. **Check transitivity:** We verify if whenever $r_{ij} = 1$ and $r_{jk} = 1$, then $r_{ik} = 1$. - For $i=1$, $j=2$, $k=3$: $r_{12} = 1$, $r_{23} = 1$, so $r_{13}$ must be $1$ (it is). - For $i=1$, $j=3$, $k=4$: $r_{13} = 1$, $r_{34} = 1$, so $r_{14}$ must be $1$ (it is). - For $i=2$, $j=3$, $k=4$: $r_{23} = 1$, $r_{34} = 1$, so $r_{24}$ must be $1$ (it is). - For $i=1$, $j=4$, $k=4$: $r_{14} = 1$, $r_{44} = 1$, so $r_{14}$ must be $1$ (it is). - Similarly for other combinations, the condition holds. Therefore, $R$ is transitive. 6. **Conclusion:** Since $R$ is reflexive, anti-symmetric, and transitive, the correct choice is: **a. All of these**