| № |
Condition |
free/or 0.5$ |
| m97526 | Using inverters, AND gates, and OR gates, construct the gates shown in Fig. 15.6. |
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| m97527 | Using only elementary row or elementary column operations and Theorems 3.2, and 3.6 (do not expand the determinants), verify the following:
(a)
(b) |
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| m97528 | Using only NAND1† gates (see Fig. 15.6), construct the inverter, AND gate, and OR gate. |
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| m97531 | Using the concept of flow in a transport network, construct a directed multigraph G = (V, E), with V = {u, v, w, x, y} and id(u) = 1, od(u) = 3; id(v) = 3, od(v) = 3; id(w) = 3, od(w) = 4; id(x) = 5, od(x) = 4; and id(y) = 4, od(y) = 2. |
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| m97535 | Using the finite state machine of Example 6.17, find the output for each of the following input strings
and determine the last internal state in the transition process. (Assume that we always start at s0.
(a) x = 1010101
(b) x = 1001001
(c) x = 101 |
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| m97536 | Using the inner product (p, q) = ∫10p(x)q(x) dx on P2, write v as the sum of a vector in U and a vector in UL.
(a) v = x2, LJ = span{x + 1, 9x - 5}
(b) v = x2 + 1, U = span{ 1, 2x - 1} |
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| m97537 | Using the laws of set theory, simplify each of the following:
(a) A ∩ (B - A)
(b) (A ∩ B) ∪ (A ∩ B ∩ ∩ D) ∪ ( ∩ B)
(c) (A - B) U (A ∩ B)
(d) |
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| m97538 | Using the result of Theorem 8.2, prove that the number of ways we can place s different objects in n distinct containers with m containers each containing exactly r of the objects is |
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| m97547 | Using the weights 2, 3, 5, 10, 10, show that the height of a Huffman tree for a given set of weights is not unique. How would you modify the algorithm so as to always produce a Huffman tree of minimal height for the given weights? |
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| m97548 | Using Venn diagrams, investigate the truth or falsity of each of the following, for sets A, B, C ⊂ U.
(a) A ∆ (B ∩ C) = (A ∆ B) ∩ (A ∆ C)
(b) A - (B U C) = (A - B) ∩ (A - C)
(c) A ∆ (B ∆ C) = (A ∆ B)∆C |
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| m97549 | Verify that (1 - x - x2 - x3 - x4 - x5 - x6)-1 is the generating function for the number of ways the sum n, where n e N, can be obtained when a single die is rolled an arbitrary number of times. |
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| m97550 | Verify that each of the cross products u x v in Exercise 1 is orthogonal to both u and v.
In Exercise 1
(a) u = 2i + 3j + 4k, v = -i + 3j - k
(b) u = i + k, v = 2i + 3j - k
(b) u = i - j + 2k, v = 3i - 4j + k
(d) |
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| m97551 | Verify that each of the following is a logical implication by showing that it is impossible for the conclusion to have the truth value 0 while the hypothesis has the truth value 1.
(a) (p ∧ q) → p
(b) p (p ∨ q)
(c) [(p ∨ q)∧ ¬ p] → q
(d) [(p → q) ∧ (r → s) ∧ (p ∨ r)] → (q ∨ s)
(e) [(p → q) ∧ (r → s) ∧ (¬ q ∨ ¬ s)] → (¬p ∨ ¬r) |
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| m97552 | Verify that for C = circ(c1, c2, c3), CTC = CCT. |
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| m97553 | Verify that for each integer n ≥ 1, |
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| m97554 | Verify that
for n = 5 and m = 2, 3, 4. |
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| m97555 | Verify that f(x) = 2x + 1 is a unit in Z4[x]. Does this contradict the result of Exercise 14? |
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| m97557 | Verify that [(p ↔ q) ∧ (q ↔ r) ∧ (r ↔ p)] ⇔ [(p → q) ∧ (q → r) ∧ (r → p)], for primitive statements p, q, and r. |
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| m97558 | Verify that [p → (q → r)] → [(p → q) → (p → r)] is a tautology. |
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| m97559 | Verify that the conclusion in Example 11.16 is unchanged if Fig. 11.48(b) has edge {a, c} drawn in the exterior of the pentagon. |
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