36 |
Chapter 3 |
We then have
A = (B + C), B + AD, D = (B + C).
Using the identity
X = Y = X + Y,
we see that the output of the product of the following sentences must be 1:
(A + B + C)(B + AD)(D + B + C).
After multiplying the above product and simplifying, we obtain
B + CAD.
So, either Etienne is the murderer, or the following events occurred simultaneously: Franc¸ois lied, Franc¸ois was not drunk and the murder took place after midnight. But Maigret knows that AC = 0, thus it follows that E = 1, i.e., Etienne is the murderer.
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146 |
Problem Construct the truth table for ( p = q) q. |
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147 |
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148 |
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n N m N n > 3 = (n + 7)2 > 49 + m |
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149 |
Problem Explain whether the following assertion is true and negate it without using the negation symbol ¬: |
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n N m N n2 > 4n = 2n > 2m + 10 |
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150Problem Prove by means of set inclusion that (A B) ∩C = (A ∩C) (B ∩C).
151Problem Obtain a sum of products for the truth table
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152 Problem Use the Inclusion-Exclusion Principle to determine how many integers in the set {1, 2, . . . , 200} are neither divisible by 3 nor 7 but are divisible by 11.
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146
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147 |
The desired truth table is |
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148 |
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The assertion is true. We have |
(n + 7)2 > 49 + m n2 + 14n > m. |
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Hence, taking m = n2 + 14n −1 for instance (or any smaller number), will make the assertion true. |
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150 |
We have, |
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(x A ∩C) (x B ∩C) |
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which establishes the equality. |
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x (A ∩C) (B ∩C), |
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