a. Let f: R" →→ R, and a € R". Prove that the following two statements are equivalent: • Du[f] (a) = 0 for all u € R"; and • Vf(a) = 0.

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: 1. Let u, ER" prove that lũ ơi <ll||||||

A: we want to shoe that  u→·v→≤u→v→

Q: For a function f: A → B and subsets C and D of A, prove that f(CUD) = f(C) Uƒ(D).

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Q: 7. Let u be transcendental over F and F< EC F(u). Show that u is algebraic over E.

A: Algebraic field

Q: 1. (a) Consider the set S = {n(-1)" : n e N}. Verify the existence or otherwise of Min(S) and…

A: a) The given set is,S=n-1n:n∈ℕ

Q: For S in the domain of f, let f ( 5 ) = { f(x): * € S}. Let C and D f de subsets of the domain of f.…

A: S is in the domain of f fS=fx: x∈S Let C and D be subsets of the domain of f Then we have to show…

Q: State the contrapositive of "If f: R→ R is strictly decreasing, then it is injective." Prove or give…

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Q: R R be given by f(x)=ax+ b %3D

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Q: Let A = R - {1} and define f: A→A by f(x) = (a) Find fo f(x). (b) Find fofo f(x). (c) Find f-¹(x).…

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Q: Q. AfX-Y where X,Y € Z* , and 0<X <23 and, 0s Ys59, X is hour of clock and Y is minute of digital…

A: Given: f: X→Y , where X is hour of clock and Y is minute hand of clock It can be clearly observed…

Q: Show that For all z E C, with z # 0, then: |z/²z=1 = z

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Q: 4. Let f: Z Z be defined by f(n) = 3n-1. Determine whether this function is 1-to-1 and/or onto.

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Q: (5) Consider sets A and B with |A| = sand |B| = t, s≥ t≥1. (a) How many onto functions are there…

A: Sets, A and B with A=s and B=t, s≥t≥1.

Q: Consider sets A and B with |A|=8 and |B|=5. How many functions f: A→B are surjective?

A: Given: The number of elements in the set A is 8 and the number of elements in the set B is 5.

Q: (9) Let f: A → B, g: B → C and h: C → D. Prove that if hogof: A→D is bijective, then f must be…

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Q: Let f: A → B be defined by f(x) = x² - 4x +5. Let A = {x|x E R and x ≥ 2} and B = {yly E R and y ≥…

A: Given thatf : AB be defined asf(x) = x2-4x+5WhereA = { x | xand x  } = [ 2 , )B = { y | yand y1 } =…

Q: Let C = {x ∈ P(A) : ∃y ∈ x, y ∈ A and y /∈ B}. Prove that P(A) − P(B) = C.

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Q: 5. Let f: DC be a function, and let A, B C D and E, FCC. Prove or disprove the following: રી..…

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Q: 3. Show that the function p : F → Ę which maps o to o is a bijection from F to Fe, where and ar to…

A: Let ϕ:F1→F2 be a function such that, ϕ0=0. Also given that ϕαr=β3r, where 0≤r≤6 When r=0,…

Q: If U-f() where f is an arbitrary function and P-U,.Q-U, then which of the following relations is a…

A: Given that : U=fxy Calculating P: P=Ux=f'xy1y ----&gt;(1) Calculating Q: Q=Uy=f'xy.x.-1y2…

Q: t K = F(u), u algebraic over F and degree of u odd. Show that K = F(u²).

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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: f. Is the domain of f(x.y)= In (6x + 5y“) bounded or unbounded? O Unbounded O Bounded

A: the domain will be Unbounded because

Q: 1. Let X be a set and f: X X be a function. Suppose that fofof is an injection. Prove that f must be…

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Q: Find the image of un der w- 2く

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Q: 6. Let z E C. Let C be set of points z satisfying Im(z) 0 and Im Prove that C = {ZEC : |Z| = 1}.…

A: Given that  Im1+z+z21-z+z2=0 We know that if Imz=0 then Im1z=0 Therefore  Im1-z+z21+z+z2=0 Finding…

Q: 3. Let f be entire. Suppose that there exist a EC and > 0 such that f(z) & Da, e) for all z E C.…

A: Since f is entire, it is holomorphic on the entire complex plane. Now, we know that f has a root in…

Q: Suppose f: R→→ R is convex. Show that for all x € [a, b]. x-a :) ≤ b = = f(a) + ² = f(b) b a f(x) ≤

A: Suppose   is convex. We have to show that:

Q: Define F: Z × Z→Z×Z as follows: For all ordered pairs (a, b) of integers, F(a, b) = (2a + 1, 3b –…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 2b. In R show that : || x - y || || x + y || ≤ 11x11 ² + 11y11²2² -(1

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Q: 4. Define f:R → R, define f (x) = 3x + 5 for all x e R. Show that a. fis one - to - one; b. f maps R…

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Q: a) Supposef:A→N is a function such that for every k∈N, there are exactly two elements x in A such…

A: Let  f:A→ℕ be a function. For every k∈ℕ, there are exactly two elements x in A such that f(x)=k. For…

Q: 5. Let f: Z→ Z be defined by f(x) = 2x-1. Determine whether f(x) is onto and/or 1 to 1. Explain your…

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Q: - Let A = {1, 2, 3, .., 103. Consider f:P(A) → N given by f(B) = |B|. That is, f takes B subset of A…

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Q: 2. If a, b E K are algebraic over F are of degree m, n, respectively, with (m, n) = 1, show that…

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Q: 3. Suppose f: A → B is injective and g: B→ A satisies f g = wherei is the identity function on B.…

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Q: 4. Determine whether each of the following functions is a bijection and if not bijection, 1 explain…

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