Relations Class 11

📝 Relations 

Ordered Pair

➤ Two numbers `a` and `b` listed in a specific order and enclosed in parentheses, form an ordered pair `(a,b)`. 

In ordered pair `(a,b)`, we call `a` as first member (or first component) and `b` as second member (or second component).

In general `(a, b) \ne (b, a)`

Example: `(2, 3) \ne (3, 2)` 

Equality of two ordered pairs

➤ We have `(a, b) = (c, d) ⟺ a = c` and `b = d`

Cartesian Products of Sets

➤ Let `A` and `B` be two non-empty sets. Then, the Cartesian product of `A` and `B` is the set denoted by `(A × B)`, consisting of all ordered pairs `(a, b)` such that `a \in A` and `b \in B`.

`∴ A × B = {(a, b): a \in A and b \in B}` 

If `A = phi` or `B = phi`, we define `A × B = phi`

Remarks

1. If `n(A) = p`  and  `n(B) = q`,  then  `n(A × B) = pq and n(B × A) = pq`.
2. If at least one of `A` and `B` is infinite then `(A× B)` is infinite and `(B × A)` is infinite. 

Relation

Consider the two sets P = {a, b, c}` and `Q = {Aman, Bhanu, Binoy, Chandra, Divya}. The cartesian product of P and Q has 15 ordered pairs which can be listed as P × Q = {(a, Aman), (a,Bhanu), (a, Binoy), ..., (c, Divya)}.

We can now obtain a subset of P × Q by introducing a relation R between the first element x and the second element y of each ordered pair (x, y) as R= { (x,y): x is the first letter of the name y, `x \in P`, `y \in Q}`. Then R = {(a, Aman), (b, Bhanu), (b, Binoy), (c, Chandra)}. A visual representation of this relation R (called an arrow diagram) is shown in Fig. 

Relation 

➤ A relation `R` from a non-empty set `A` to a non-empty set `B` is a subset of the cartesian product `A × B`. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in `A × B`. The second element is called the image of the first element.

`R \subseteq A × B`

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`. 

 `dom(R) \subseteq A `

Range 

➤ The set of all second elements in a relation `R` from a set `A` to a set `B` is called the range of the relation `R`, written as Range `(R)`.  The whole set `B` is called the codomain of the relation `R`. Note that range `\subseteq` codomain. 

Work in progress.

References

1. NCERT Mathematics Class XI
2.  Mathematics (R.S. Aggarwal) Class XI
3. Mathematics (R.D. Sharma) Class XI