Let $A=\left\{1,2,3\right\}.$ Then number of equivalence relations containing $\left(1,2\right)$ is

An ordered pair consists of two objects or elements in a given fixed order.

2. Equality of Two Ordered Pairs:

$\left(a_1,b_1\right)=\left(a_2,b_2\right)\iff a_1=a_2$ and b1=b2

3. Cartesian Product of Two or Three Sets:

(i) If $A$ and $B$ are two non-empty sets, then $A\times B=\left\{\left(a,b\right):a\in A,b\in B\right\}$ is called the Cartesian product of $A$ and $B.$

(ii) If $A$ and $B$ are finite sets having $m$ and $n$ elements respectively, then $A\times B$ has $mn$ elements.

(iii) $R\times R=\left\{\left(x,y\right):x,y\in R\right\}$ is the set of all points in $XY$-plane.

(iv) $R\times R\times R=\left\{\left(x,y,z\right):x,y,z\in R\right\}$ is the set of all points in three-dimensional space.

4. Some Properties of Cartesian Product:

For any three sets $A,B,C$ we have

(i) Distributive property over set union:

$A\times \left(B\cup C\right)=\left(A\times B\right)\cup \left(A\times C\right)$

(ii) Distributive property over set intersection:

$A\times \left(B\cap C\right)=\left(A\times B\right)\cap \left(A\times C\right)$

(iii) Distributive property over set difference:

$A\times \left(B-C\right)=\left(A\times B\right)-\left(A\times C\right)$

(iv) $A\times B=B\times A\iff A=B$

(v) $\left(A\times B\right)\cap \left(B\times A\right)=\left(A\cap B\right)\times \left(B\cap A\right)$

(vi) $A\times \left(B\text{'}\cup C\text{'}\right)\text{'}=\left(A\times B\right)\cap \left(A\times C\right)$

(vii) $A\times \left(B\text{'}\cap C\text{'}\right)\text{'}=\left(A\times B\right)\cup \left(A\times C\right)$

(viii) $A\times B=A\times C\Rightarrow B=C$

5. Relation:

Let $A$ and $B$ be two sets. A relation from $A$ to $B$ is a subset of $A\times B.$

6. Number Of Relations:

If $A$ and $B$ are finite sets having $m$ and $n$ elements respectively. Then, $2^{mn}$ relations can be defined from $A$ to $B.$

7. Domain and Range Of a Relation:

If $R$ is a relation from set $A$ to set $B,$ then Domain $\left(R\right)=\left\{x:\left(x,y\right)\in R\right\},$ Range $\left(R\right)=\left\{y:\left(x,y\right)\in R\right\}$

8. A relation from a set $A$ to itself is called a relation on $A.$

9. Inverse of Relation:

Let $A,B$ be two sets and let $R$ be a relation from set $A$ to set $B.$ Then the inverse of $R,$ denoted by $R^{-1}$ is a relation from $B$ to $A$ and is defined by $R^{-1}=\left\{\left(b,a\right):\left(a,b\right)\in R\right\}$.

Clearly, $\left(a,b\right)\in R\iff \left(b,a\right)\in R^{-1}$

Domain $\left(R\right)=$ Range $\left(R^{-1}\right)$, and Range $\left(R\right)=$ Domain $\left(R^{-1}\right)$

10. Functions:

Let $A$ and $B$ be two non-empty sets. Then a relation $f$ from $A$ to $B$ is a function, if

(i) For each $a\in A$ there exists $b\in B$ such that $\left(a,b\right)\in f$

(ii) $\left(a,b\right)\in f$ and $\left(a,c\right)\in f\Rightarrow b=c$.

In other words, $f$ is a function from $A$ to $B$ if each element of $A$ appears in some ordered pair in $f$ and no two ordered pairs in $f$ have the same first element.

If $\left(a,b\right)\in f,$ then $b$ is called the image of a under $f$.

(iii) A function $f$ from a set $A$ to a set $B$ is a rule associating elements of set $A$ to elements of set $B$ such that every element in set $A$ is associated to a unique element in set $B$.

Here the set $A$ is called the domain of $f$ and the set $B$ is called its co-domain.

(iv) The range of a function $f$ is the set of images of elements in the domain.

11. Real-Valued Function:

A real function has the domain and co-domain both as subsets of set $R$.

12. Algebra of Real Functions:

If $f:D_1\to R$ and $g:D_2\to R$ are two real functions and $c\in R,$ then

(i) Addition or Subtraction of Real Functions:

$f\pm g:D_1\cap D_2\to R$ is defined as $\left(f\pm g\right)\left(x\right)=f\left(x\right)\pm g\left(x\right)$

(ii) Multiplication of Real Functions:

$fg:D_1\cap D_2\to R$ is defined as $\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)$

(iii) Quotient of Real Functions:

$\frac{f}{g}:D_1\cap D_2-\left\{x:g\left(x\right)=0\right\}\to R$ is defined as $\left(\frac{f}{g}\right) \left(x\right) \mathrm{ }=\mathrm{ }\mathrm{ }\frac{f\left(x\right)}{g\mathrm{ }\left(x\right)}$

(iv) Multiplication by a scalar:

$cf:D_1\to R$ is defined as $\left(cf\right)\left(x\right)=cf\left(x\right).$