In Exercises 21 − 42 , define the necessary symbols, rewrite the argument in symbolic form, and use a truth table to determine whether the argument is valid. If the argument is invalid, interpret the specific circumstances that cause the argument to be invalid. 1. Drinking espresso is sufficient for not sleeping. 2. Not eating dessert is necessary for being on a diet. 3. Not eating dessert is sufficient for drinking espresso. Therefore, not being on a diet is necessary for sleeping.
In Exercises 21 − 42 , define the necessary symbols, rewrite the argument in symbolic form, and use a truth table to determine whether the argument is valid. If the argument is invalid, interpret the specific circumstances that cause the argument to be invalid. 1. Drinking espresso is sufficient for not sleeping. 2. Not eating dessert is necessary for being on a diet. 3. Not eating dessert is sufficient for drinking espresso. Therefore, not being on a diet is necessary for sleeping.
In Exercises
21
−
42
, define the necessary symbols, rewrite the argument in symbolic form, and use a truth table to determine whether the argument is valid. If the argument is invalid, interpret the specific circumstances that cause the argument to be invalid.
1. Drinking espresso is sufficient for not sleeping.
2. Not eating dessert is necessary for being on a diet.
3. Not eating dessert is sufficient for drinking espresso.
Therefore, not being on a diet is necessary for sleeping.
Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p-1)/2
multiple of n, i.e.
n mod p, 2n mod p, ...,
2
p-1
-n mod p.
Let T be the subset of S consisting of those residues which exceed p/2.
Find the set T, and hence compute the Legendre symbol (7|23).
The first 11 multiples of 7 reduced mod 23 are
7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8.
23
The set T is the subset of these residues exceeding
2°
So T = {12, 14, 17, 19, 21}.
By Gauss' lemma (Apostol Theorem 9.6),
(7|23) = (−1)|T| = (−1)5 = −1.
how come?
Shading a Venn diagram with 3 sets: Unions, intersections, and...
The Venn diagram shows sets A, B, C, and the universal set U.
Shade (CUA)' n B on the Venn diagram.
U
Explanation
Check
A-
B
Q Search
田
3. A different 7-Eleven has a bank of slurpee fountain heads. Their available flavors are as follows: Mountain
Dew, Mountain Dew Code Red, Grape, Pepsi and Mountain Dew Livewire. You fill five different cups full
with each type of flavor. How many different ways can you arrange the cups in a line if exactly two Mountain
Dew flavors are next to each other?
3.2.1
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