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: 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: 1. f¹ B-A. 2. f is bijective. Suppose f: A→ B. Prove that the following statements are equivalent.

A: A bijection, also known as a bijective function, one-to-one correspondence, or invertible function,…

Q: 2. Let A be a 2 x 2 matrix. Assume ₁ and ₂ are the two distinct non-zero eigenvalues of A. Determine…

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

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Q: #1 same holds for all x, y E R. (First prove that a" ay at+y for all x,y E Q, prove that the at+y…

A: We know, The supremum of a subset of a partially ordered set is the least element in that is greater…

Q: 6. Define h(r) = for all r. Prove that the function h: R→ R satisfies the € - 6 criterion on R.

A: In this question, we want to prove that the given function satisfy the given criteria on R.

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: Let f: R -> R and E ⊆ R. Prove f is continous iff f-1(E) is closed whenever E ⊆ R is closed

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Q: 4. Define f: [0, 1] → R by if x E Qn [0, 1] if x ¢ Qn [0, 1] f(x) = (a) Prove that S° f = 0; s*f =…

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Q: Q1 Not Both Let f: Z Z where f(x) = r. Then f is one-to-one, but not onto. %3D O True O False

A: f is one to one and onto because foe every distinct integer x, x3 is also distinct and vice versa.

Q: 2. Let f: A → B and g: C → D be functions. The product of f and g is the function defined as…

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Q: 23. Let A be a non-empty set of positive numbers, which is bounded above and let B = {xe R: 1/xe A}.…

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Q: Let E = {0, 2, 4, 6, ...}, define ƒ: E × E → Z¹ by the formula f(m, n) = 4m5¹, \ (m, n) € Ɛ × Ɛ.…

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

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

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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: f: R→ R and g: Z → Z by the rules f(x) g(n) = n² for all n € Z. (i) Is f onto? Prove or give a…

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Q: ·f is defined as f: R – {3}→ R -, R is the set of real numbers. x+5 f(x) 2х-6 . Is f one-to-one?…

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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: ® let the functions: 4:R-A gR-R,with こXe2 Prove that: Tx(4g) =4Vxg +CVf)xg

A: Given two functions: f(x,y,z)=x2eyz3, g(x,y,z)=(x2z2,-xyz,y3ez) To prove  ∇×(fg)=f∇×g+(∇f)×g

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: 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: 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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Q: Let X and Y be sets,and supposet hat f:X→Y,g:Y→X, and h:Y →X are functions such that g◦f=1X…

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Q: Let f:Z → Z be given by f (x) = 3x + 1. a. Show that f is one-to-one. 7. b. Is f onto? EXPLAIN.

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Q: 3. Consider the set X = {a,b,c} and the function f: X→ X defined by the rules f(a) = b. f (b) = a…

A: The set X=a, b, c and the function f:X→X defined by the rules fa=b, fb=a, fc=b. We have to find the…

Q: Define f: Z→ Z as f(n) = an + b, where a, b E Z. Find the necessary and sufficient contitions on a…

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Q: 2. Give an example of a domain U and predicates P and Q such that is true but is false. (Vxe U,…

A: 2. Given that  a domain U and predicates P and Q such that                    ∀x∈U, P(x)→(∀x∈U,…

Q: 8. (p f(A) ≤ f(B), then ACB. Let f: XY. Prove that for all A, B C X, if f is one-to-one and

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Q: Let b, c E R and b>1+c. Show that the following equation has two distinct real solutions: r2 + bx +c…

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