Definitions [8]
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.
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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.
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In general, (a,b) ≠ (b,a)(a,b).
If A and B are two non-empty sets, then their Cartesian product is written as \[A \times B\].
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It is the set of all ordered pairs \[(a, b)\] such that \[a \in A\] and \[b \in B\].
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\[A \times B = \{(a, b) : a \in A, b \in B\}\].
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
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every element of the domain must be used
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no element of the domain can have more than one image
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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.
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Codomain: If R is a relation from A to B, then the set B is called the co–domain of the relation R.
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Range: The set of all second components of all ordered pairs in a relation R is called the range of the relation.
f: X → Y is a function if each element of X is associated with a unique element of Y
- Domain (X): Set of all input values
- Codomain (Y): Set of all possible outputs
- Range: Set of actual output values of f
- Range ⊆ Codomain
Theorems and Laws [1]
Prove that the greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
f : R → R, given by f(x) = [x]
It is seen that f(1.2) = [1.2] = 1 and f(1.9) = [1.9] = 1.
∴ f(1.2) = f(1.9), but 1.2 ≠ 1.9.
∴ f is not one-one.
Now, consider 0.7 ∈ R.
It is known that f(x) = [x] is always an integer. Thus, there does not exist any element x ∈ R such that f(x) = 0.7.
∴ f is not onto.
Hence, the greatest integer function is neither one-one nor onto.
Key Points
| Term | Meaning |
|---|---|
| Ordered Pair | Pair of elements written in a fixed order |
| Cartesian Product | Set of all ordered pairs from two sets |
| Relation | Subset of a Cartesian product |
| Domain | Set of first elements of a relation/function |
| Codomain | Target set into which mapping occurs |
| Range | Actual set of output values obtained |
| Function | Relation assigning exactly one output to each input |
| Type of Function | Condition | Key Idea |
|---|---|---|
| One-One (Injective) | f(x₁) = f(x₂) ⇒ x₁ = x₂ | Different inputs → different outputs |
| Onto (Surjective) | Range = Codomain | Every element of the codomain is mapped |
| Into Function | Range ⊂ Codomain | Some elements of the codomain are not mapped |
| Many-One Function | x₁ ≠ x₂ but f(x₁) = f(x₂) | Different inputs → same output |
| Bijective Function | One-one + Onto | Perfect mapping (1-1 and onto) |
