Let X = {1,2,3,4} and Y = {5,6,7}. State whether a function f: X → Y can be one-to-one or not without justification. Also, produce an example of a function f: X →Y that is not onto.

Q: Determine whether or not the function f : Z × (Z − {0}) → Z is onto, if f((m, n)) = ⌊m/n ⌋. (Use…

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Q: Let A, B, C be sets. Let f : A → B and g : B → C be functions. a) Prove that if f and g are…

A: We shall solve both parts of this question that you have posted.

Q: Theorem 4. Let X be a set, and let F: X Y be a l-1 and onto function. (a) F-lo F = Idx (b) FoF- =…

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Q: Are the following functions 1-1, onto, both, or neither. f: R-R. f(x) = x13 h: Z- Z. h(x) = x g:…

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Q: Let A={a,b,c,d,e}. Find a relation R on A which is symmetric and transitive but not reflexive

A: Reflexive relation means every term of a set is related with itself

Q: 15. Determine whether the function f: Z x Z Z is onto if a) f(m, n) =m+n. b) f(m, n) = m² + n². c)…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: 1. Let X = {m, a, t, h}, Y = {1,4, 0}, Z = {d,i, s, c, r, e,t, e} (a) Define a function f : X →Y…

A: Since you have posted a question with multiple sub-parts, we will solve the first three subparts for…

Q: b) A function g:Z→Z is defined by the rule g(m, n) = m + n Determine whether it is onto or not...

A: A function f:A→B is said to be onto or surjective if for all element b∈B there is an element a∈A…

Q: 1. Let f : Z6 → Z6 be such that f(a) = x2 + 3. (a) Is f a well defined function? (b) Is f…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub parts for…

Q: Suppose h is a function such that h(3) = -2, h'(3) = 2, h"(3) = 4, h(5) = 6, h'(5) = 7, h"(5) = 13,…

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Q: Let f : {1, 2, 3} → {a, b, c} be one-one and onto function given by f(1) = a,   f(2) = b and f(3) =…

A: Consider

Q: Let function f: Z Z be defined as follows; 1 x is odd f(x) = . x² + 2 x is even The functionf is .…

A: To show that  a function f(x) is one-one , we have to show that for all x , y belong to the given…

Q: Let f, g,h: A→ A be any bijective functions. Then (f ogoh)-

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Q: Let f :A→B and g : B +C be functions. Show that if g o f is onto, then g is onto.

A: Given: f:A→B and g:B→C are functions Given: gof: A→C is onto To show: g:B→C is onto That is,  for…

Q: Let A and B be sets and f a function from A to B. Then the following assertions hold: f'is onto if…

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Q: 5. Determine whether the function f : Z × Z →Z is onto if a) f (m, n) = m² + n² b)f (m, n) = m³

A: See the solution.

Q: A function f() from {a, b, c, d} to { 1, 2, 3, 4} defined by f(a) = 3, f(b) = 2 and f(c) = 1 and…

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Q: Exercise 2.12. Determine whether or not the given functions f:Z Z are one-to-one or onto (or neither…

A: The solution of given functions is given as follows :

Q: (a) Prove that there exists a function f : Z → Z so that f is one-to-one but f is not onto. (b)…

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Q: Is the function f(x) = x + 1 from the set of integers to the set of integers onto?

A: For onto from set A to B, Every value of set B should exist atleast once in the function.

Q: Let f be the function from {a, b, c, d} to {1, 2, 3, 4} with f(a) = 4, f(b) = 2, f(c) = 1, and f(d)…

A: For bijection, the function needs to be one to one and onto.

Q: 5. Given a function f: A → B. Define the relation f from P(A) to P(B) by f = {(X,Y)C P(A) × P(B) | Y…

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Q: Prove that for functions f : R→R, the e- 6 definition of continuity implies the open set definition.

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Q: A function g: Z → Z is defined by the rule g(m,n)=m+n Determine whether it is onto or not...

A: Onto functions: A function f: A→B is said to be an onto function if range of function is equal to…

Q: Show that if f and g are continuous on [a, b] and if f(a) g (b), then there exists some ce ]a, b[…

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Q: Prove or provide a connter C are functions such that gois one-to-one. then f is one-to-one. xample…

A: See the detailed solution below.

Q: Let f : A → B and g : B C be surjective functions. Prove or disprove whether gof is surjective.

A: Here given that                       f :A→B   and    g :B→C  be both surjective function .  Now we…

Q: a. The function f : Z → Z, f(x) = 2x is onto. (A) True (В) False b. The function f : Z → Z, f(x) =…

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Q: Prove the theorem: For finite sets X and Y, let |X|=m and |Y|=n. If there is an onto function f:X →…

A: please see the next step for solution

Q: Let f : Z → Z be relation defined by f(x) = ||æ| – 5|+ x for all æ E Z. i) f is a function ii) f is…

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Q: Let X = {1,2,3,4} and Y = {5,6,7}. State whether a function f : X → Y can be one-to-one or not…

A: By using definition of one-time and onto function we solve the given problem :

Q: 7. If a functionf defined in [a, b] is such that o (i) f is derivable in [a, b], (ii) f' (a) and f'…

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Q: For each of the functions below, indicate whether the function is onto, one-to-one, neither or both.…

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Q: Let A and B be nonempty sets. Prove the following statements. (a) For any function f : A → A, if f o…

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Q: Let g: A ---------- B andf: B-----------C. Prove that f is onto if f 0 g is onto.

A: Given:g:A→B;     f:B→CTo Check : if (f∘g):A→C then f is ontoNow given…

Q: (a) Prove that there exists a function f: Z → Z so that f is one-to-one but f is not onto. (b) Prove…

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Q: Let A and B be sets. Show that               f : A × B → B × A such that                  f(a, b) =…

A: Given:

Q: Is the function f(x) = x2 from the set of integers to the set of integers onto?

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Q: Let A be a nonempty set and let f : A → B be a function. Prove that f is one-to-one if and only if…

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Q: Show that a one-one function                 f : {1, 2, 3} → {1, 2, 3} must be onto.

A: Given, the function f : {1, 2, 3} → {1, 2, 3} is one-one.

Q: Prove or provide a counterexample to the following statement: if f : A → B and g : B → C are…

A: i have used the definition of onto functions and showed that c has no preimage under f  but gof is…

Q: Given g, a function from A to B and f, a function from B to C. "If f and g are onto, then f • g is…

A: Given that g:A→B is onto. If y∈B, then there exists a pre-image x∈A. ⇒g(x)=y

Q: Let function f:Z Z be defined as follows; x– 1 x is odd f(x) = < (x² + 2 x is even The function f is…

A: Let, x=2 that is x is even. Then, fx=x2+2 So, f2=22+2=6f-2=-22+2=6 Therefore, f2=f-2 Then, the given…

Q: Let A and B be nonempty sets. Prove the following statements. (a) For any function f:A→A, if f◦f is…

A: (a) Let z be an element of A. Since fof is onto there is an element x in A such that fof(x)=z.…

Q: Let V: U → R be a strict Lyapunov function, where U CR" is a neighborhood of a compact and invariant…

A: Let V:U→R be a strict Lyapunov function, where U⊂ℝn is a neighborhood of a compact and invariant set…