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Functions (TYPES OF FUNCTIONS (Bijective (One-to-One Onto) Functions (is…
Functions
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EQUAL FUNCTIONS
consider two functions f and h from a set X to a set Y. The functions f and h are called equal functions if and only if f(a) = h(a), for every a ∈ X.
The functions f and h are called unequal functions if there exist at least one element a ∈ X such that f(a) ≠ h(a).
A function f from a set P into a set Q is a relation from P to Q such that each element of P is related to exactly one element of the set Q
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FUNCTIONS AS A SET
If P and Q are any two non-empty sets, then a function f from P to Q is a subset of P × Q, with two important restrictions
∀ a ∈ P, (a, b) ∈ f for some b ∈ Q
If (a, b) ∈ f and (a, c) ∈ f then b = c.
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IMAGE OF AN ELEMENT
f the element x of P corresponds to y under function f, then y is the image of x under f and is written as
If f(x) = y, then we say that x is a per image of y.
If f: X → Y, then each element of P has unique image in Q, whereas every element in Q need not be image of some x in P.
RANGE OF A FUNCTION
The range of a function is the set of images of its domain. In other words, we can say it is a subset of its co-domain. It is denoted as f(domain).
If f: P → Q, then f(P) = {f(x): x ∈ P} = {y: y ∈ Q | ∃ x ∈ P, such that f(x) = y}.
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COMPOSITION OF FUNCTIONS
Consider functions, f: A → B and g: B → C. The composition of f with g is a function from A into C defined by (gof) (x) = g[f(x)] and is denoted by gof.
To find the composition of f and g, first find the image of x under f and then find the image of f(x) under g.
PERMUTATION FUNCTIONS
permutation is the act of arranging the members of a set into a sequence or order, or, if the set is already ordered, rearranging (reordering) its elements—a process called permuting.
Inverse of a Permutation
The inverse of a permutation (which is written in matrix form) can be obtained by exchanging the two rows and rearranging the columns so that the two row is in order.
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Disjoint Cycles
Two cycles are called disjoint cycles of a given set A if there does not exist an element a ∈ A such that a appears in both cycles.
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