Given two non-empty sets \(A\) and \(B\). The Cartesian product \(A \times B\) is
the set of all ordered pairs of elements from \(A\) and \(B\), i.e.,
\(A \times B = \left\{ {\left( {a,b} \right):a \in A,b \in B} \right\}\)
If either \(A\) or \(B\) is the null set, then \(A \times B\) will also be
empty set, i.e., \(A \times B\)

2. What is relations?

A relation \(R\) from a non-empty set \(A\) to a non-empty set \(B\) is a subset of the cartesian product \(A \times B\).
It "maps" elements of one set to another set. The subset is derived by describing a relationship
between the first element and the second element of the ordered pair \(\left( {A \times B} \right)\). Domain: The set of all first elements of the ordered pairs in a relation \(R\) from a set
\(A\) to a set \(B\) is called the domain of the relation \(R\). Range: the set of all the ending points is called the range

3. What is Function

A function is a "well-behaved" relation

A function \(f\) is a relation from a non-empty set \(A\) to a non-empty set \(B\) such that the domain of \(f\) is \(A\) and no
two distinct ordered pairs in \(f\) have the same first element.

For a relation to be a function, there must be only and exactly one \(y\) that corresponds to a given \(x\)

If \(f\) is a function from \(A\) to \(B\) and \(\left( {a,{\rm{ }}b} \right) \in f\), then \(f\left( a \right) = b\), where \(b\) is called the image of \(a\) under \(f\) and \(a\) is called the preimage of \(b\) under\(f\).

4. Algebra of Real Function

Real Value Function: A function which has all real number or subset of the real number as it domain Real Valued Function: A function which has all real number or subset of the real number as it range
For functions \(f:{\rm{ }}X - > {\bf{R}}\) and \(g:{\rm{ }}X - > {\bf{R}}\), we have

Addition \(\left( {f + g} \right)\left( x \right) = f\left( x \right) + g\left( x \right),x \in X\)

Substraction \(\left( {f - g} \right)\left( x \right) = f\left( x \right)-g\left( x \right),x \in X\)

Multiplication \(\left( {f.g} \right)\left( x \right) = f\left( x \right).g\left( x \right),x \in X\)

Multiplication by real number
\(\left( {kf} \right)\left( x \right) = k{\rm{ }}f\left( x \right),x \in X\), where \(k\) is a real number.

Division \(\frac{f}{g}\left( x \right) = \frac{{f(x)}}{{g(x)}}\)
\(x \in X\) and \(g\left( x \right) \ne 0\)

Please click here to get the full material for Class X1