Revision: Sets and Relations Maths HSC Commerce (English Medium) 11th Standard Maharashtra State Board

If A and B are two non-empty sets, then their Cartesian product is written as \[A \times B\].

• It is the set of all ordered pairs \[(a, b)\] such that \[a \in A\] and \[b \in B\].

• \[A \times B = \{(a, b) : a \in A, b \in B\}\].

An ordered pair is a pair of objects whose components occur in a special order. It is written by listing the two components in the specified order, separating them by a comma and enclosing the pair in parentheses. In the ordered pair (a, b), a is called the first component and b the second component.

• Two ordered pairs are equal only if their corresponding components are equal, so (a,b) = (c,d) if and only if a = c and b = d.

• In general, (a,b) ≠ (b,a).

A relation from set A to set B is any subset of the Cartesian product \[A \times B\].

So, if \[R \subseteq A \times B\], then R is a relation from A to B.

A function from set A to set B is a relation in which every element of A has exactly one image in B.

Condition for a Function

• every element of the domain must be used

• no element of the domain can have more than one image

• different elements may have the same image

• Domain: The set of all first components of the ordered pairs in a relation R is called the domain of the relation R.

• Codomain: If R is a relation from A to B, then the set B is called the co–domain of the relation R.

• Range: The set of all second components of all ordered pairs in a relation R is called the range of the relation.