23. 2.6. a) Give an example of a function f : N ---> N which is injective but not surjective. A not-injective function has a “collision” in its range. A non-injective non-surjective function (also not a bijection) . f(x) = 0 if x ≤ 0 = x/2 if x > 0 & x is even = -(x+1)/2 if x > 0 & x is odd. (v) f (x) = x 3. 4. It is seen that for x, y ∈ Z, f (x) = f (y) ⇒ x 3 = y 3 ⇒ x = y ∴ f is injective. A function f : A + B, that is neither injective nor surjective. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. Note that is not surjective because, for example, the vector cannot be obtained as a linear combination of the first two vectors of the standard basis (hence there is at least one element of the codomain that does not belong to the range of ). Proof. Injective and surjective are not quite "opposites", since functions are DIRECTED, the domain and co-domain play asymmetrical roles (this is quite different than relations, which in a sense are more "balanced"). b) Give an example of a function f : N--->N which is surjective but not injective. 21. 22. c) Give an example of two bijections f,g : N--->N such that f g ≠ g f. Thus when we show a function is not injective it is enough to nd an example of two di erent elements in the domain that have the same image. 2. Prove that the function f: N !N be de ned by f(n) = n2, is not surjective. 6. Let the extended function be f. For our example let f(x) = 0 if x is a negative integer. Give an example of a function F :Z → Z which is injective but not surjective. A function is a way of matching all members of a set A to a set B. Give an example of a function F:Z → Z which is surjective but not injective. Whatever we do the extended function will be a surjective one but not injective. Now, 2 ∈ Z. Example 2.6.1. Injective, Surjective, and Bijective tells us about how a function behaves. ∴ f is not surjective. Hence, function f is injective but not surjective. Thus, the map is injective. This relation is a function. But, there does not exist any element. Hope this will be helpful The number 3 is an element of the codomain, N. However, 3 is not the square of any integer. Then, at last we get our required function as f : Z → Z given by. Example 2.6.1. It is injective (any pair of distinct elements of the … Give an example of a function … A function f : BR that is injective. It is not injective, since \(f\left( c \right) = f\left( b \right) = 0,\) but \(b \ne c.\) It is also not surjective, because there is no preimage for the element \(3 \in B.\) The relation is a function. f(x) = 10*sin(x) + x is surjective, in that every real number is an f value (for one or more x's), but it's not injective, as the f values are repeated for different x's since the curve oscillates faster than it rises. A function f :Z → A that is surjective. x in domain Z such that f (x) = x 3 = 2 ∴ f is not surjective. There is an important quality about injective functions that becomes apparent in this example, and that is important for us in defining an injective function rigorously. A function f : B → B that is bijective and satisfies f(x) + f(y) for all X,Y E B Also: 5. explain why there is no injective function f:R → B. 3. An example of a function behaves us about how a function f: --! That f ( x ) = x 3 = 2 ∴ f is not surjective ned... 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