7. Let A and B be sets, and let f: A → B and g: B → A be functions. Suppose that: • go f(a) = a for every a € A; ● fog(b) = b for every b € B. Prove that f is bijective.

Q: Given the 5 relations below, select all that are not function. Remark: If A is a set, then A is…

A: Given data:- R= x,y : x ∈ ℤ ∧ y ∈ ℤ ∧ y2 = x .R= x,y : x ∈ Pℤ ∧ y ∈ ℤ ∧ y ∈ x .R= x,y : x ∈ Pℤ ∧ y ∈…

Q: Consider sets A and B with |A| = 7 and |B| = 4. How many functions f: A → B are surjective?

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Q: Let X be a non-empty set. Define the function h : X → P(X), x → {x}. Which of the following is true?…

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Q: Give a proof of the following statement. Suppose A, B and C are sets, f is an injective function…

A: We have to prove the given statement.

Q: Let A = {1, 3, 5, 6, 7, 9, 10, 12} and B = {1, 2, 3, 5, 7, 8, 9, 11, 12}. Consider the following…

A: Given a function f: A -> B , we have to determine whether it is injection, surjection,…

Q: 3. Let f: A → B and g: B → C be functions. Recall that the composition of g and f is the function go…

A: a) To prove that if g and f are injective, then go f is injective, we need to show that for any two…

Q: Let A and B be two sets and f be function. f: A B.Assuming the function f is invertible, what is the…

A:

Q: 1.3.2. Let f: X → Y be a function, let B be a subset of Y, and let {B;}ie1 be a family of subsets of…

A: Let f:X→Y be a function, let B be a subset of Y and Bii∈I be a family of subsets of Y.  To prove…

Q: b) Let the set A = {-1,0, 1} and consider the following function: f2 A → A given by f₂(x) = (-x)4.…

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Q: A function f : Z Z is defined by f(n) = 3n + 1. Let E be the set of even %3D integers. Determine the…

A: When a function is linear, it is one-one and onto function. We know that a bijective function is…

Q: Let A, B be sets and f : A → B a function. Let A' CA and B' C B. (a) Prove that f¯¹[ƒ[A']] 2 A'.…

A:

Q: Let A = {a, b, c}. Define the function f: p(A)→N, by f(X) = card(X), where p(A) be the power set of…

A: Given A = {1,2,3} then ρ(A) = {ϕ , {a} , {b} , {c} , {a,b} , {a,c} , {b,c} , {a,b,c}} Define f :…

Q: ? ? ? Hint: ✰✰✰✰✰✰✰.. 5. {(b, 6), (h, 2), (0, 6), (r, 4), (s, 27)} 6. {(b, 2), (h, 27), (0, 4), (r,…

A: For the given set , does it define function f:X-->Y .

Q: 2. Let R be the relation on the set A = (1, 2, 3, 4, 5, 6, 7} defined by the rule (a,b) e R if the…

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Q: How many functions f: {1,2,3,4} {1,2,3,4} are there satisfying: a. f(1) = 1 or f(2)= 2 (or both)? b.…

A: Solution :-

Q: Let A, B and C be sets, and let g: A->B and f:B‐>C be functions. Prove if f, g are surjective, then…

A: The solution is given as

Q: Let g : B → C and h : B → C be functions. Suppose that go f = hof for every function f: A → B. Prove…

A: We have to prove that g(x) = h(x) for all x belongs to B.

Q: 3. Let f: A B and g B → A be functions such that gof = i4. Prove that if g is injective, then f is…

A:

Q: Let A, B and C be arbitrary sets, and let g : A → B and f : B → C be functions. Show:  if f, g are…

A:

Q: Let f : A → B be a function. Prove that for any B1, B2, ⊂ B, f -1(B1) ∪ f -1(B2) ⊂ f -1(B1 ∪ B2)

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Q: Let (A,) be a poset. Define a function f: A → P(A) by f(x) = {a € A: a <x}. What can you say about x…

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 = {3, 5, 7, 8} and define a relation S from A to A as follows: %3D For all (x, y) E A × A, (x,…

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Q: : 4. Let f A → B be a bijective function. Construct a function g : B → A such that g(f(a)) = a for…

A: Given that, f:A→B be a bijective function. We have to construct a function g:B→A such that gfa=a for…

Q: (a) Give an example of two functions f: A→B and g: B-C such that f is both 1-1 and onto but gof is…

A: (a)let's define sets 𝐴, 𝐵, and 𝐶 as follows:Let 𝐴={1,2}Let 𝐵={3,4}Let 𝐶={5}Now, let's define…

Q: Give a proof of the following statement. Suppose A, B and C are sets, f is a surjective function…

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Q: 2. Let f A → B be a function. Prove that if g, h : B → A are inverse functions of f, then g(b) =…

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Q: Let A, B and C be sets, and let g: A → B and f: B → C be functions. Prove: (i) If f, g are…

A: (i) Let, f : B → C and g: A → B are injective.      Now, f ◦ g is a function from A to C.…

Q: Prove the statement. Let f : A → B . If X ⊆ A, Y ⊆ B , and f-1[Y] = {x ∈ A|f(x) ∈ Y}, then f[X] ∩ Y…

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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: The Power Set a) Let A = {a, b, c}. Find P(A). b) Let B = {0, 1} Let f: P(A) → B. The input of f is…

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Q: How many functions f: {1,2,3,4} → a. ƒ(1) = 1 or ƒ(2) = 2 (or both)? → {1, 2, 3, 4} are there…

A: Since your question carry more than three subparts , so as per Bartleby guidelines we are answering…

Q: Let A = Z and B = Z. Suppose f: A ∈ B is defined by the rule f(a) = a3 + a + 1. Determine f(−1) and…

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Q: Suppose that A and B are sets and f : A –→ B is a function. Fix subsets X,Y C A. Prove that Imf(X) –…

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Q: 3. Let ƒ : A → B be a function. Prove that if ƒ has an inverse, then fis bijective.

A: This question is related to topic function from real analysis.

Q: a. Prove that for any sets A,B⊆X and for any function f:X⟶Y, f(A∩B)⊆f(A)∩f(B).       b. Prove that…

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Q: Domain of F Co-domain of F X Y F a • d• Domain of G Co-domain of G X Y G a • b e of d• a. Is F…

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Q: Exercise 4.4.5. Let A and B be sets. Prove that there is a bijective function f: A × B → BXA.

A: A and B are two sets.  We need to prove there is a bijective function            f : A × B → B × A.

Q: Let A, B and C be nonempty sets and let f : A → B, g: BC and h: B → C be functions. For each of the…

A: