[PDF] Functions Surjective/Injective/Bijective PDF cemtl

Understand what is meant by surjective, injective and bijective, • Check if a function has the above properties Surjective Functions Let f : A → B be an arbitrary

f is bijective if it is surjective and injective (one-to-one and onto) Discussion We begin by discussing three very important properties functions defined above 1 A

[PDF] Functions Surjective/Injective/Bijective

Understand what is meant by surjective, injective and bijective, • Check if a function has the above properties Surjective Functions Let f : A → B be an arbitrary

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3 The map f is bijective if it is both injective and surjective Lemma 1 2 Let f : A → B be a function Then the following are true i) Function f is injective iff f−1({b})

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We say that f is a bijection if f is both injective and surjective It is interesting to go through the examples above The function in (1) is neither injective or surjective

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Injective Functions ○ A function f : A → B is called injective (or one-to-one) if We will prove that the function g ∘ f : A → C is also surjective To do so, we will

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Une fonction g est dite injective si et seulement si tout réel de l'image Une fonction h est dite bijective si et seulement si elle est et injective et surjective

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A few words about notation: To define a specific function one must define the domain, the codomain, and the rule of correspondence In other words, f : A → B

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Properties of Functions: Surjective • Three properties: surjective (onto), injective, bijective • Let f: S → T be an arbitrary function – every member of S has an

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[PDF] Functions Surjective/Injective/Bijective

Functions

Surjective/Injective/Bijective

Aim To introduce and explain the following properties of functions: \surjective", \injective" and \bijective".

Learning Outcomes

At the end of this section you will be able to:

²Understand what is meant by surjective, injective and bijective,

²Check if a function has the above properties.

Surjective Functions

Letf:A!Bbe an arbitrary function with domainAand codomainB. Part of the de¯nition of a function is that every member ofAhas an image underfand that all the images are members ofB; the setRof all such images is called the range of the function f. ThusR=f(A) and clearlyRµB. If it should happen thatR=B, that is, that the range coincides with the codomain, then the function is called asurjectivefunction. De¯nition :A functionf:A!Bis ansurjective, oronto, function if the range of fequals the codomain off. In every function with rangeRand codomainB,RµB. To prove that a given function is surjective, we must show thatBµR; then it will be true thatR=B. We must therefore show that an arbitrary member of the codomain is a member of the range, that is, that it is the image of some member of the domain. On the other hand, if we can produce one member of the codomain that is not the image of any member of the domain, then we have proved that the function is not surjective. To show that a function is surjective pick an arbitrary element in the codomain and show that it has a preimage in the domain. Graph the following function and check is it surjective? f:R! fxjx >0g; f(x) =ex

Functions

The codomain isx >0. By looking at the graph of the functionf(x) =exwe can see thatf(x) exists for all non-negative values, i.e. for all values ofx >0. Hence the range of the function isx >0. This means that the codomain and the range are identical and so the function is surjective. Graphically speaking, if it is possible to draw a horizontal line across the graph of a function without making contact with the curve representing the function then the function is not surjective.

Graph the following two functions

f:R!R; f(x) =x3; f:R!R; f(x) =x2: and check to see if they are surjective. The answers are (1) yes, (2) no. Can you see why?

Injective Functions

The de¯nition of a function guarantees a unique image of every member of the domain. A given member of the range may have more that one preimage, however. If this is the case then the function is notinjective. De¯nition :A functionf:A!Bisinjective, orone-to-one, if no member ofBis the image underfof two distinct elements ofA. To show that a function is injective, we assume that there are elementsa1anda2of

Awithf(a1) =f(a2) and then show thata1=a2.

Graphically speaking, if a horizontal line cuts the curve representing the function at most once then the function is injective. Test the following functions to see if they are injective. f:R!R; f(x) =x3; f:R!R; f(x) =x2; f: [0;1)!R; f(x) =x2;

Functions

Solutions:

Injective

Not Injective

Injective

Bijective Function

De¯nition :A functionf:A!Bisbijective(abijection) if it is bothsurjective andinjective. Iff:A!Bis injective and surjective, thenfis called aone-to-one correspondence betweenAandB. This terminology comes from the fact that each element ofAwill then correspond to a unique element ofBand visa versa. Which of the following functions are surjective, injective and bijective ? f:R!R; f(x) =x3; f:R!R; f(x) = 2x; f:R!R; f(x) =x3¡2x2¡5x+ 6;

[PDF] [PDF] Functions Surjective/Injective/Bijective

[PDF] 2 Properties of Functions 21 Injections, Surjections - FSU Math