6. Let A = {x|x € R and x² - 4x + 3 < 0} and B = {x|x € R and 0 < x < 6}. Prove that AC B.

Q: Show that if x > 0, then 1 + 1x − x² ≤ √1+x≤ 1 + ½ x. -

A:

Q: Suppose x is a positive rational number and x? < 2. Let y = 22. Prove that x < y and y? < 2. X+2

A: We will use basic knowledge of set theory and logic to answer this question.

Q: 2. Find p and q such that 7x+1 5 |5x-1|≤4 implies p ≤: <q

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Q: . Prove that an→L = Jan] → |L]. (Make cas

A: Given that an is a sequence which converges to L that is, an→L. Then we have to prove that |an|…

Q: Find the supremum and infimum of the set x € R:x = +(-1)" ,n e N} + (-1)" ,n e N}

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Q: XA and x & f(x)}. Show Cim ƒ.) 38. Let a, b, c, and d be any real numbers such that a < b and c < d.…

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Q: 5. Let F= [3,2,1] and r= [2,1,–3]. Compute the following: %3D a) Proj „F. b) Orth rF = F – Proj „F.…

A: We know that  ProjrF=r.F∥r∥2r Now, r.F=23+12-31=5 ∥r∥2=r,r        =2,1,-3,2,1,-3        =22+11-3-3…

Q: n 5. Let n be a fixed positive integer. Denote by Po the set of all real polynomials of degree n or…

A: A nonempty set V together with two operations - vector addition and scalar multiplication is called…

Q: Prove or disprove that ⌈x + y⌉ = ⌈x⌉ + ⌈y⌉ for all real numbers x and y.

A: According to the given information, It is required to prove or disprove that:

Q: 5) Suppose that W1 < R" and W2 4 R". Prove (W1 n W2) < R"

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Q: Prove that if x is a real number such that x ≥ 0 and x ≤ € for all € E R where € > 0, then x = = 0.

A: This question is from real analysis.Answer given below.

Q: Show that if x,y,z,w are real numbers such that x < y and z ≤w, then x + z < y + w.

A:

Q: 6. Let ao, a1, a2, a3, a4 be constant real numbers such that a1 a2 a3 + + + + a4 = 0. 5 4 3 2 Show…

A: Our objective is given below:

Q: Suppose that x is a negative real number and that x < 1/x. Prove that x < 1.

A: Step 1: Step 2: Step 3: Step 4:

Q: Z is the set of integers, R is the set of real numbers. 1) Functions f, g, h are defined as f: R ->…

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Q: 2. Show that if a]b then a'|b² for all integers a and b (Symbols V, 3, A, V, V,A,→,-)

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Q: 10. Prove the following inequaltities for any positive real numbers x,y: xy³≤ x² + y4 (ii) xy³+x³y…

A: Given Data: We have to prove the following inequaltities for any positive real numbers x,y : i…

Q: Prove the following statement: (Va € Z) (Vb Z) (Vc = Z) [((a | b) ^ (a | (b+c))) ⇒ (a | 3c)]

A: