Let f be a function from X to Y. For A CX, let also f(A) denote the set f(A) = {f(a) | a E A}. Prove that f is one-to-one if and only if f(An B) = f(A)n f(B) for any subsets A, BC X.

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Q: Let A and B be subsets of a set X. Prove that if A ⊆ Bc, then A and B are disjoint.

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Q: Let A = {-4, 0, 4} and B = {t, u, v, w}. Define a function F: A → B by the following arrow diagram:…

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Q: Let A = {1,2,3}, B B = = Define the functions f: AC and g: B → C by f(1) = 7, f(2)= 9, f(3) = 7,…

A: We want g o h = f.     So, we want g o h(1) = f(1) = 7            and g o h(3) = f(3) = 7   Now,…

Q: Consider f = {(x, y) : x ∈ N and x + y = 5}. Write out some elements of f. What are the domain and…

A: Given: f={(x,y): x∈N and x+y=5} The domain of a function f(x) is the set of all values for which the…

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A: In the given question we have to find f and g are one - one function.

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A: Here given that be a poset.Define function is  by Noted that here  is power set of  .Here we have to…

Q: Let A = {1,2,...,n} and consider the functions f: A → A. a. How many functions satisfy f(1) = 1? b.…

A: This question related to function and bijection.

Q: Prove that if f: R→ R is continuous, then the image of a bounded set over f is bounded.

A: As per the question we have to prove that if any function f : ℝ → ℝ is continuous, then the image of…

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A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: Let A = {a, b, c, d, e}, B = {1, 2, 3, 4, 5} and let us consider the function f : A → B, f(a) = 3,…

A: Given that:  f : A → B, f(a) = 3, f(b) = 4, f(c) = 2, f(d) = 5, f(e) = 1.   Since, the images (i.e.…

Q: For any set X with at least 2 elements, define a function f : P(X) x P(X) → P(X) by f(A,B) = A △B…

A: We have taken two different ordered pair of subsets of X and shown that their image is same. Hence f…

Q: True or False A function is a relation between two sets Dand R so that each element x in the first…

A: By definition: A function is a relation from a set of inputs to a set of possible outputs where each…

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

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Q: Let f be the subset of Z × Z defined by  f = {(ab, a + b) : a, b ∈ Z}. Is f a function from Z to Z?…

A: Let f be the subset of Z × Z defined by  f = {(ab, a + b) : a, b ∈ Z}. We observe that  That is the…

Q: 3. Let A be a finite set. In this exertise functions f, g will be of type g: A →A and f: A → A. (a)…

A: Injective and surjective function

Q: Show that the point of intersection of two lines in the plane of afield F lies in the plane of F.

A: Point of intersection:- The junction of two lines is referred to as the point of intersection. The…

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Q: Let f: A → B and g: B → C be two injective functions. Prove that go f is injective.

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Q: Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined byf(x) = ax + b, for some…

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