Q.Prove that the relation R defined by $R = \{(a, b) : a \le b\}$ on the set R of real numbers is reflexive and transitive but not symmetric.

Check each property of $R=\{(1,1),(2,2),(1,2)\}$ on $A=\{1,2,3\}$.

Reflexive? Requires $(a,a)\in R$ for every $a\in A$, i.e. $(1,1),(2,2),(3,3)$. Since $3\in A$ but $(3,3)\notin R$, $R$ is not reflexive.

Symmetric? Requires $(b,a)\in R$ whenever $(a,b)\in R$. Here $(1,2)\in R$ but $(2,1)\notin R$, so $R$ is not symmetric.

Transitive? Requires $(a,c)\in R$ whenever $(a,b),(b,c)\in R$. The only linking pairs are: …

To determine the properties of the given relation $R$ on the set $A = \{1, 2, 3\}$, we need to check if it satisfies the definitions of reflexivity, symmetry, and transitivity. Understanding these definitions precisely is key to avoiding common errors.

A relation $R$ on a set $A$ is:

• Reflexive if for every element $a \in A$, the ordered pair $(a, a)$ is in $R$. This means every element must be related to itself.
• Symmetric if for every pair $(a, b) \in R$, the pair $(b, a)$ is also in $R$. This means if $a$ is related to $b$, then $b$ must also be related to $a$.
• Transitive if for every $a, b, c \in A$, whenever $(a, b) \in R$ and $(b, c) \in R$, it must follow that $(a, c) \in R$. This means if $a$ is related to $b$ and $b$ is related to $c$, then $a$ must also be related to $c$.

Let's examine $R = \{(1, 2), (2, 1), (1, 1)\}$ on the set $A = \{1, 2, 3\}$ step by step.

1. Check for Reflexivity:

   For $R$ to be reflexive, every element in $A$ must be related to itself. That is, $(1, 1)$, $(2, 2)$, and $(3, 3)$ must all be present in $R$.

   • We see that $(1, 1) \in R$.
   • However, $(2, 2) \notin R$.
   • Also, $(3, 3) \notin R$. Since $(2, 2)$ and $(3, 3)$ are not in $R$, the relation $R$ is not reflexive.

2. Check for Symmetry:

   For $R$ to be symmetric, for every pair $(a, b) \in R$, the reverse pair $(b, a)$ must also be in $R$.

   Let's check each pair in $R$:

   • For $(1, 2) \in R$, we need to check if $(2, 1) \in R$. Yes, $(2, 1) \in R$.
   • For $(2, 1) \in R$, we need to check if $(1, 2) \in R$. Yes, $(1, 2) \in R$.
   • For $(1, 1) \in R$, we need to check if $(1, 1) \in R$. Yes, $(1, 1) \in R$. Since for every ordered pair $(a, b)$ in $R$, the pair $(b, a)$ is also in $R$, the relation $R$ is symmetric.

3. Check for Transitivity:

   For $R$ to be transitive, if $(a, b) \in R$ and $(b, c) \in R$, then $(a, c)$ must also be in $R$. We need to check all such combinations.

   • Consider $(1, 2) \in R$ and $(2, 1) \in R$. Here, $a=1, b=2, c=1$. According to the definition of transitivity, $(a, c) = (1, 1)$ must be in $R$. We see that $(1, 1) \in R$. This case holds.
   • Consider $(2, 1) \in R$ and $(1, 2) \in R$. Here, $a=2, b=1, c=2$. …