CBSE Class 12 Mathematics

## Relations and their representation The **Cartesian product** $A \times B$ of two sets is the set of all ordered pairs $(a, b)$ with $a \in A$ and $b \in B$. A **relation** $R$ from $A$ to $B$ is any subset of $A \times B$, and we write $a\,R\,b$ when $(a, b) \in R$. A relation **on** a set $A$ is a relation from $A$ to $A$, so it is a subset of $A \times A$. If $A$ has $n$ elements then $A \times A$ has $n^2$ ordered pairs, and each pair is either in a relation or out of it, so the number of relations on $A$ is $2^{n^2}$. For $A = \{1, 2\}$ that is $2^4 = 16$. Two extreme relations are named. The **empty relation** has no pairs at all, so $R = \varnothing$. The **universal relation** contains every pair, so $R = A \times A$. A relation on a small set can be written as a list of pairs, drawn as arrows between the elements, or read from a rule such as "$a\,R\,b$ if $a - b$ is even". ## Reflexive, symmetric and transitive relations A relation $R$ on a set $A$ is - **reflexive** if $(a, a) \in R$ for every $a \in A$; - **symmetric** if $(a, b) \in R$ implies $(b, a) \in R$; - **transitive** if $(a, b) \in R$ and $(b, c) \in R$ together imply $(a, c) \in R$. Each property is a statement about all elements, so one failing case is enough to show that it does not hold, and to show that it does hold you must argue for a general element. A single missing pair decides reflexivity: on $\{1, 2, 3\}$ the relation must contain $(1,1)$, $(2,2)$ and $(3,3)$. For a relation given as a…