Exercise 4. Prove that if f : A → B and g : B → C are bijective, then (gof) : A→C is bijective. then r 3 for all intogomo Exercise 5, Prove that if T. = 1 and r

Q: Let R= {(a, b) : a <b} Prove or disprove that R is transitive.

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Q: Let f : R R with f(x) = -2x +5 Prove that f is onto.

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Q: Suppose v1, V2,..., Vm E V. Prove that {v1, v2, ..., Um} = (span(v1, v2, ..., Um))-.

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Q: Let ((u,,u2,U3),(V1,V2,V3))= 2u, V1 + 3u2V2+ U3V3 be an inner product on R. Determine ||(1, – 1, -…

A: Explanation of the answer is as follows

Q: Show that X is Hausdorff if and only if the diagonal △ = {(x, x)|x ∈ X} is closed in X × X.

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Q: Consider the set V = {(x, y) = R² : x+y=1}. on this set defined in the following way: (x2, y2) = (x₁…

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Q: assume that Pr[A]=0.25Pr[A]=0.25, Pr[B]=0.35Pr[B]=0.35, and Pr[A∩B]=0.1Pr[A∩B]=0.1. 1)Pr[A|B] =…

A: Given: P(A) =0.25 P(B)=0.35 P(A∩B) =0.1

Q: Let vị = V2 = V3 V4 and -[1} Confirm that p = V1 +v2 + ¿v3 + ¿V4 and vị – V2 + V3 – V4 = 0. Jse the…

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Q: prove or disprove: a) The function f: ℤ × ℤ → ℤ defined by f(m, n) = m + n + 1 is onto. b) The

A: (a) the function f is onto. (b) the function f is not one-to-one.   The detailed solution is as…

Q: = Prove the operation of addition on the set Q = {[(x, y)] : x ≤ Z, y ≤ N} defined by [(x, y)] +…

A: Let us show that given operation is well defined.

Q: Show that Z[V-2] is a Euclidean Domain with respect to the norm N defined by N(a + by-2) = a² + 2b².…

A: This is a problem of abstract algebra.

Q: Forz € C, define the linear operator T₂ : C → C by T₂(w) = W. Characterize those z for which Tz,…

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Q: prove that two vectur ae orthognal A = x + 4y+32 B =4x+2y-42

A: See the attachment

Q: and n 12/22 -42 Z+6z Z722+2 2ナ z2, 4z+3

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Q: Exercise 4.4.5. Let A and B be sets. Prove that there is a bijective function f: A × B → BXA.

A: A and B are two sets.  We need to prove there is a bijective function            f : A × B → B × A.

Q: 7

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Q: Let f = u + iv be analytic in the connected open set and assume u and v satisfy u? + 3v2 = 4. Show f…

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Q: Exercise 3. Prove that a relation R on A is antisymmetric if and only if RNR-' C {(a, a) | a E A

A: We know that , A relation R on set A is antisymmetric  if either x , y∉ R  or (y , x) ∉ R whenever…

Q: :N N amd 9: N-oN are functions such that f is not onto but f+9 is onto Prove that 9 is onto or give…

A: Suppose that f:ℕ→ℕ and g:ℕ→ℕ are functions such that f is not onto but f+g is onto. To prove that g…