If X and Y are finite sets, prove that XY = |X|Y. Here, XY is the set of all functions from Y to X.

Q: Let A, B, C denote sets. Let g, h : A → B and f : B → C denote functions. Assume f is one-to-one and…

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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  X = {a, b, c}  and  Y = {d, e, f, g}.   Define functions H and K by the arrow diagrams below.…

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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: Let ‘E’ & ‘F’ are any two non-empty sets and there order are as follows;…

A: given let ‘E’ & ‘F’ are any two non-empty sets and there order are as follows;…

Q: Find a set A and a relation R such that the relation is symmetric, and not reflexive and transitive.

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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: 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 D, E be non-empty sets, let ACD, let BCE, and let f: D E be a function. Prove that Anf¹(B) ≤…

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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…

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Q: On sets, the relation x R y given by there exists a bijective function from the set x to the set y'…

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Q: Let F = [−2, 3) ∪ {−3, π}. Determine max F, inf F,  interior of F, boundary of F,  and closure of F.

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Q: A = {1, 3} and B = {2, 4, 6}. Define a relation R from A to B as follows: for all (x, y) ∈ A × B…

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Q: B. Let S and T be sets of real numbers, and define S. T to be {x.y:x e Sandy € T}. Show that if S…

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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: 8. Let A = {1,2} and B = {a, b, c}. For each of the following cases, decide whether the set…

A: The set A={1,2}, B=a,b,c & f:A→B. a) 1,a,2,b b) 1,b,2,b c) 1,a,1,b,2,c

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: Let X and Y be sets with A,B ⊆ X. If f : X →Y is a function, prove that f(A∪B) =f(A)∪f(B)

A: I am going to solve the problem by using some simple algebra to get the required result of the given…

Q: For a set A, let {0,1} A be the set of all functions from A to the two point set {0,1}. Prove that…

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Q: Let R be a relation on Z given by aRb if and only if a = b². Is this relation a function? Justify…

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Q: Use two of the following sets for each part below. Let X = {a,b,c}, Y = {1, 2, 3, 4} and Z = {s, t,…

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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: If X, Y are finite sets with |X|=4>2= |Y. How many surjective functions f: X→ Y are there? 2 16 8 O…

A: Let us solve above problem

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: A is a non-empty set. Show that the power set of A is in one to one correspondence with the set of…

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Q: Let X be the set {5, –6, 2}. For the first three parts of this problem you are asked to define a…

A: Consider the set 5,-6,2. The function is f:X→X and the relation is u~w⇔w=fu.   a) The relation is…

Q: Determine whether the function f from { a, b, c, d} to { a, b, c, d} is injective (one - to - one),…

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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…

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Q: Consider sets A and B with |A| = 12 and |B|| 8. How many functions f : A → B are surjective?

A: We will find out the required value.

Q: Let A, B, and C be nonempty sets and let f: A -> B, g: B -> C and h: B -> C be functions. Find a…

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Q: True or false: There exists a monotonic function f: R → R such that the set of points of…

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Q: (ii) Let A be a non-empty set and f: AA be a function. Prove that if fof is injective, then / is

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Q: . For nonempty sets A, B and C, let f : A → B and g: B → C be functions. (a) Prove: If g of is…

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Q: Show that for every pair of sets X and Y, the sets X × Y and U {(x, y) | y ≤ Y} xEX are equal.

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