9 t Suppose fe R[a, b] and m < f(x) < M, for all æ € [a, b). Prove: m(b – a) < f(x) dar < M(b – a).

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: 3.23 Prove or give a counterexample: The function |f| € R[a, b] + ƒ € R{a, b].

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Q: 4. Let F = {s+t√2 s,te Q}. Show that F is a subfield of R. You may assume that √2&Q.

A: The objective of the question is to prove that the set F, defined as F = {s + t * sqrt(2) | s, t…

Q: Let G = D8 = (r, s | r8 = id = s², sr = r²s). : (a) (b) (c) Let N = (4). Show that N ≤G. Find the…

A: Solution:-  (a) To show that N is a normal subgroup of G, we need to verify that for any n in N and…

Q: 7. Let A = fa; b; c; dg and B = fx; yg. (a) How many functions are there from A to B? How many of…

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Q: Show that A = (aj) = Cmxn. Then || A||2 ≤ || A||F ≤ √ Rank(A)|| A||2 V

A: Matrix A=(aij)∈Cm×n: This notation indicates that A is a matrix with complex entries, where m and n…

Q: Let Z∼N(0,1). Find P(-2<Z<1).

A: From the provided information,Z~N (0, 1)The probability P (-2&lt;Z&lt;1) need to be calculated.

Q: d) Let the set A = {1,2,3,4} and let P(A) denote the power-set of A. Consider the following…

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Q: 5. Let f : R -→ R be defined by f(x) = 2x +9. Find a formula for f-1.

A: We need to find the inverse function of the real valued function f(x)=2x+9

Q: 2. Let f: ZxZ-ZxZ be defined by f(m, n) = (3m+n,n²) a. Is fone-to-one? b. Is fonto?

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Q: d) Let the set A = {1, 2, 3,4} and let P(A) denote the power-set of A. Consider the following…

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Q: 8. Let f:A→B be a function. Use the principles of logic to give useful denials (not counterexamples)…

A: Given: f:A→B be a function. To do: Use the principles of logic to give useful denials of: a)…

Q: 1. Let D = the sales of DVDs in billions of dollars, S = the total sales of streaming videos in…

A: The sales of dvds given as the function D = -0.92t + 8.75 and the total sales of streaming videos…

Q: Prove: If f1, f2 € R(a) on [a, b] with fi < f2, then ' fi da < s' fz da

A:

Q: f: A → B is a function, Let B1 ⊆ B ve B2 ⊆ B. f−1(B1 ∪ B2) = f−1(B1) ∪ f-1(B2) Show that it is…

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Q: By using the fact that P[AU B] 1 –Ei=, P[A£].

A: (a). Prove that P [A U B U C] ≤ P[A] + P[B] + P[C]: It is given that P [A U B] ≤ P[A] + P[B]. The…

Q: What is true about the function -- asx-m? OAM-f B) (x) 0 from below OC) f(x) is undefined OD)x) from…

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Q: Consider the function vec : Cmxn → Cmn such that for any A E Cmxn A*1 vec (A) = = A*2 : A*n Prove…

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Q: Let B = R − {3}. Define g : B → B by g(t) = 3t/(t − 3) Show that the function g is surjective.…

A: We are given that B = R − {3}. Define g : B → B by g(t) = 3t/(t − 3). We need to show that g is…

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…

A: