Let S = {c, o, u, n, t, m, e, 2}. Observe |S| = 8. (a) How many total functions are there f:S → S if there are no restrictions? (b) How many partial functions are there f:S → S if there are no restrictions? (c) How many total functions are there f: S→ S with restrictions that f(m) is a vowel and f(u) is not a vowel? Recall the vowels are a, e, i, o and u.

Q: Let f: R → R be defined by f(x) = 4x² + 2 and set y = f(x). (a) The level sets of f are subsets of…

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Q: Let f(z) be an entire function satisfying that f(z)| ≤|z² for all z. Show that f(z) = az² for some…

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Q: 5. Let X be the set of odd integers and Y = {a € Z | a is even and a ≤ 20}, and consider the…

A: (a) Let, f(x) = f(y)          => x(9 – x) = y(9 – y)          => 9x - 9y - x2 + y2 = 0…

Q: 4. Suppose P is the set of people alive today and C is the setof possible eye colors. Let f : P → C…

A: Given :          P is the set of people alive today and C is the set of possible eye colors.…

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: 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: Let g : B → N × N where BCZ given by g(m) = (|m| + 2, |m| + 3). Choose all subsets B such that g is…

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Q: Given that there exists a surjective function f : S → T, which of the following substitutions for…

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Q: X = {p, q, r, s} Y = {4, 5, 6, 7, 8} The function f: X → Y is defined as: f = {(s, 8), (r, 8), (q,…

A: We want to find elements in the range of function f.

Q: Let f : N → P(N) be the function defined by f(n) = {n, n + 1, . . . , 2n} The following statement…

A: Given a mapping f where f:ℕ→Pℕ defined by  fn=n,n+1,n+2,...2n It can be seen that the set fn…

Q: 1. Show that if f [a, b] → R is increasing, then f is of bounded : variation.

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Q: Evaluate fff f(x, y, z) dV for the specified and B: B f(x, y, z): Z | 8 2 ≤ x ≤ 6,0 ≤ y ≤ 6,

A: We have to find the value of the given expression according to the given information

Q: 15. Show that if f: A- 1 f(GnH)=f(G) B and G, H are subsets of B, then f-¹(GUH)=f(G) Uf (H) and nf…

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Q: (a) Show that the formula (VæR(g(x), x) = Vx3yR(y,x)) is universally valid, where g is a 1-place…

A: Part(a): As we know, ∀xRgx,x⇒∀x∃yRy,x Where g is a two place function symbol and R is a two place…

Q: X = {p, q, r, s} Y = {4, 5, 6, 7, 8} The function f: X→ Y is defined as: f = {(r, 7), (s, 8), (p,…

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Q: Let A = {a, b, c, d} and f = { (a, 0), (c, 12), (d, 18), (b, 12) }. Is f a function from A to B ?…

A: Given, A = {a,b,c,d} f = {(a,0) , (c,12) , (d,18) , (b,12)}

Q: 1. Show that if f [a,b] → R is increasing, then f is of bounded variation.

A: Given f: a, b → ℝ is increasing function.

Q: Let X = {a, b, c, d,e} and let T = {X, Ø, {a}, {a,d}, {a,e}}, then * O (X,T) is not connected but it…

A: No information is given for topology on Y So i have solved only the middle question

Q: 11. Define F:Z × Z→Z × Z as follows: For every ordered pair (a, b) of integers, F(a, b) = (2a+1, 3Ь…

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Q: 4. Let X an Y be sets with A, B C X and let f: X→Y be a function. Recall that {f(x) EYE A}, and…

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Q: Let 1 f(x) 2 f(=) = ;= - 7, - 4 < x < 10 -1 The domain of f is the interval [A, B] where A Preview…

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Q: A Compary profit con be modeled by the function pCA=lt-sy cwhere p is the profit lin thousands of…

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Q: 18. Let A = {1,2, 3}, B = f: A → B, and g : B → C, such that {W, Y, Z}, C = {7, V2, V5}. Find…

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Q: 1. Suppose f: A B and W and X are subsets of A. (a) Will it always be true that f(W U X) = f(W) U…

A: We are given that f: A --&gt; B and W and X are subsets of A. For (a), The given statement f(W∪X) =…

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: 3. (a) Let f: X → Y and g: Y→ Z be functions. Prove the following statements. If f and g are both…

A: We can prove the following parts .

Q: Refer to the following functions g and h. g(n) = n² +n h(n) = n² –n Evaluate each function below: a)…

A: The given functions are gn=n2+nhn=n2-n

Q: If f is a function from X to Y and A, B are subsets of X then .A f(A – B) = f(A) – f(B) .B f(AUB) =…

A: Given: f:X→Y and a,b⊆X, Find out: To choose the correct option.