Question

Show that the relation R in R (set of real numbers) is defined as $R=\{\left( a,b \right):a\le b\}$ is reflexive and transitive but not symmetric.

Answer 1

Hint: Now we have that $R=\{\left( a,b \right):a\le b\}$ which means for any real number a and b $\left( a,b \right)\in R$ iff $a\le b$ . Now we have the relation is symmetric if for all real numbers $\left( a,a \right)\in R$ Now the relation is called a symmetric relation if we have for all $\left( a,b \right)\in R$ then $\left( b,a \right)\in R$ .
Now a relation is called a transitive relation if for $\left( a,b \right)\in R$ and $\left( b,c \right)\in R$ we have $\left( a,c \right)\in R$
Hence we will check all the conditions and show the required result.

Complete step-by-step answer:
Now let us consider the relation defined as $R=\{\left( a,b \right):a\le b\}$ .
Now consider any number such that a is a real number.
Now we know that a = a.
Hence we can also write that $a\le a$ .
Hence we can say that $\left( a,a \right)\in R$ .
This means relation R is symmetric.
Now let us say that a, b and c are real numbers such that $\left( a,b \right)\in R$ and $\left( b,c \right)\in R$ .
Now since we have $\left( a,b \right)\in R$ this means that $a\le b$
Now since we have $\left( b,c \right)\in R$ we can say that $b\le c$
This means we have $a\le b\le c$ . Which means nothing but $a\le c$ .
Hence we have $\left( a,c \right)\in R$ .
Hence for $\left( a,b \right)\in R$ and $\left( b,c \right)\in R$ we have $\left( a,c \right)\in R$ .
Hence we get that R is transitive.
Now let us say that $\left( a,b \right)\in R$ .
Hence we get, $a\le b$
Now we know that since \[a\le b\] we have $b\ge a$
Hence b can never be less than a.
This means $\left( b,a \right)\notin R$ .
Hence for $\left( a,b \right)\in R$ , $\left( b,a \right)\notin R$.
Hence we have the relation R is not symmetric.

Note: Now note here we have taken the variables a, b and c arbitrarily hence we can easily say that the conditions are true for all real numbers. Also the relation which is symmetric, reflexive as well as transitive is called an equivalence relation.