A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. 2. Hope this will be helpful 2.6. Proof. It is seen that for x, y ∈ Z, f (x) = f (y) ⇒ x 3 = y 3 ⇒ x = y ∴ f is injective. x in domain Z such that f (x) = x 3 = 2 ∴ f is not surjective. A function f : A + B, that is neither injective nor surjective. Give an example of a function F:Z → Z which is surjective but not injective. 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. Hence, function f is injective but not surjective. A function f : BR that is injective. 3. 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. Then, at last we get our required function as f : Z → Z given by. 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 ). A not-injective function has a “collision” in its range. Let the extended function be f. For our example let f(x) = 0 if x is a negative integer. This relation is a function. Example 2.6.1. Thus, the map is injective. 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. 22. a) Give an example of a function f : N ---> N which is injective but not surjective. A function is a way of matching all members of a set A to a set B. A non-injective non-surjective function (also not a bijection) . 4. 23. Injective, Surjective, and Bijective tells us about how a function behaves. It is injective (any pair of distinct elements of the … Prove that the function f: N !N be de ned by f(n) = n2, is not surjective. 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. 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. Whatever we do the extended function will be a surjective one but not injective. 21. 6. 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"). A function f :Z → A that is surjective. 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. Now, 2 ∈ Z. 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