| № |
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free/or 0.5$ |
| m96127 | Prove that the sum of the cubes of three consecutive integers is divisible by 9. |
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| m96128 | Prove that the sum, product, and scalar multiple of diagonal, scalar, and upper (lower) triangular matrices is diagonal, scalar, and upper (lower) triangular, respectively. |
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| m96130 | Prove that the three-by-three grid of Fig. 11.34 is isomorphic to a subgraph of the hypercube Q4. |
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| m96131 | Prove that the vector space P of all polynomials is not finite-dimensional. Suppose that {p1(t), p2(t), ( ( ( ( Pk(t)} is a finite basis for P. Let dj = degree Pj (t). Establish a contradiction. |
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| m96136 | Prove that Z+ × Z+ × Z+ = {(a, b, c) | a, b, c ∈ Z+} is countable. |
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| m96138 | Prove the following result in three ways (as in Theorem 2.4): If n is an odd integer, then n + 11 is even. |
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| m96144 | Prove the Jacobi identity (u x v) x w + (v x w) x u + (w x u) x v = 0. |
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| m96147 | Prove the parallelogram law for any two vectors in an inner product space:
|u + v||2 + ||u-v||2 = 2||u||2 + 2||v||2 |
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| m96151 | Prove Theorem 12.6 and Corollary 12.1.
Theorem 12.6
Let T = (V, E) be a complete m-ary tree with |V| = n. If T has i leaves and i internal vertices, then
(a) n = mi + 1;
(b) ℓ = (m - 1)i + 1; and
(c) i = (ℓ - 1)/(m - 1) = (n - 1)/m.
Corollary 12.1.
Let T be a balanced complete m-ary tree with i leaves. Then the height of T is ⌈logm ℓ⌉. |
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| m96152 | Prove Theorem 13.2. |
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| m96153 | Prove Theorem 17.14.
L1, L2 be an orthogonal pair of n X n Latin squares. If L1, L2 are standardized as L*1, L*2, then L*1, L*2 are orthogonal. |
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| m96154 | Prove Theorem 17.7.
For polynomials in F[x],
(a) Every nonzero polynomial of degree < 1 is irreducible.
(b) if fix) ∈ F[x] with degree f(x) = 2 or 3, then f(x) is reducible if and only if f(x) has a root in the field F. |
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| m96155 | Prove Theorem 2.
Let (e1, e2, ..., en} be an orthogonal set of vectors.
1. {r1, e1, r2e2,..., rnen} is also orthogonal for any r, ≠ 0 in R.
2. |
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| m96156 | Prove Theorem 5.9.
If f: A -> B, g: B -> C are invertible functions, then g o f: A -> C is invertible and (g 0 f)-1 = f-1 °g-1. |
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| m96157 | Prove Theorem 7.5.
Theorem 7.5
If (A, R) is a poset and B ⊆ A, then B has at most one lub (glb). |
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| m96160 | Prove Theorems 17.9 and 17.10.
Euclidean Algorithm for Polynomials let fix), g(x) ∈ F[x] with degree f(x) < degree g(x) and f(x) ≠ 0 applying the division algorithm, we write
Then rk(x), the last nonzero remainder, is a greatest common divisor of f(x), g(x), and is a constant multiple of the monic greatest common divisor of f(x), g(x). [Multiplying rk(x) by the inverse of its leading coefficient allows us to obtain the unique monic polynomial we call the greatest common divisor.]
Let s(x) ∈ F(x), s(x) 0. Define relation R on F[x] by f(x) R g(x) if f(x) - g(x) = t(x)s(x), for some t(x) ∈ F[x] - that is, six) divides f(x) - g(x). Then R is an equivalence relation on F[x]. |
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| m96162 | Provide a counterexample to show that the result in the preceding exercise is false if gcd(m, n) > 1 |
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| m96163 | Provide a proof by contradiction for the following: For every integer n, if n2 is odd, then n is odd. |
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| m96164 | Provide a proof for the opposite inclusion in Theorem 7.1.
Theorem 7.1
Let A, B, C, and D be sets with R1 ⊆ A × B, R2 ⊆ B × C, and R3 ⊆ C × D. Then R1 o (R2 o R3) = (R1 o R2) o R3. |
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| m96165 | Provide a recursive definition for each of the following languages A⊂∑* where E = {0, 1}.
(a) x ∈ A if (and only if) the number of 0 s in x is even.
(b) x ∈ A if (and only if) all of the l s in x precede all of the 0 s. |
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