Apply the fundamental Boolean operations to the following situation: "Dawson is a programmer AND Dawson knows Python."

Chuck, a meticulous chef at a popular restaurant, is known for following precise recipes. Applying De Morgan’s Laws, we can understand that if Chuck decides not to use some of the specified ingredients (¬ A) or if he opts not to follow the recipe’s instructions (¬ B), the resulting dish may not turn out as expected. Conversely, if Chuck both uses all the ingredients (A) and follows the recipe exactly (B), then he maximizes the chances of creating a delicious meal, adhering to the conjunction rule of De Morgan’s Laws. Rewrite the statements using De Morgan’s Laws and show how logical operations distribute over each other when negating either part of Chuck’s recipes.

Tisha, a compassionate volunteer at the local animal shelter, ensures that all animals receive proper care and attention. Applying De Morgan's Laws, if Tisha neglects to provide food or exercise to the shelter animals (¬ A), or if she fails to offer them affection (¬ B), it may lead to distress among the animals. Conversely, when Tisha provides both food and exercise (A) and offers affection (B), she fulfills the needs of the animals, following the conjunction rule of De Morgan's Laws.

Ava is deciding whether to go for a walk based on the weather conditions. She will create a truth table to help make this decision. Suppose she have two conditions that influence her decision:

1. Condition A: Is it sunny?

2. Condition B: Is it not raining?

And she wants to decide whether she should go for a walk based on these conditions. Let us assume that she will go for a walk only if both conditions are met. In this case, the logical expression is A AND B (both conditions must be true for Ava to go for a walk).

Use inductive reasoning to find the next term in the sequence.

1. \(5, 8, 11, 14,\, \ldots\)

2. \(\vartriangle, \triangleright, \triangledown, \triangleleft,\, \ldots \)

3. \(1, 4, 9, 16,\, \ldots\)

Each sequence shown here is an arithmetic sequence. Find the next two numbers in each sequence.

a. 10, 16, 22, . . .

b. \(\displaystyle\frac{1}{2}\), 1, \(\displaystyle\frac{3}{2}\), . . .

c. 5, 0, \(-5\), . . .

Each sequence shown here is a geometric sequence. Find the next number in each sequence.

a. 2, 10, 50, . . .

b. 3, \(-15\), 75, . . .

c. \(\displaystyle\frac{1}{8}\), \(\displaystyle\frac{1}{4}\), \(\displaystyle\frac{1}{2}\),. . .

Find the number of bees in the tenth generation of the family tree of a male honeybee.

Patterns and Connections 2:  The Fibonacci Sequence

Fibonacci introduces us to an interesting sequence of numbers that has many applications in the world around us.

Identify the hypothesis and conclusion in each statement.

a. If a and b are positive numbers, then -a(-b) = ab.

b. If it is raining, then the streets are wet.

c. C = 90° ⇒ c2 = a2 + b2

For the statement below, write the converse, the inverse, and the contrapositive.

     If it is raining, then the streets are wet.

If the statement "If I always tell the truth, then I never have to remember what I said" is true, give another statement that must also be true.

If the following statement is true, what other conditional statement must also be true?

Write the following statement in "if/then" form.

         Romeo loves Juliet.

A ski resort in Vermont advertised their new high-speed chair lift as the worlds fastest chair lift, with a speed of 1,100 feet per second. Show why the speed cannot be correct.

A snapshot appeared in USA Today in November of 2005 that shows the rate of births and deaths in the United States. There is one birth every seven seconds. How many births are there in one week?

The pacific plate is moving \(519\) km every 5 million years. What is the rate of movement in cm/tear?

Pyroclastic flows travel at speeds of 10 meters/second to 100 meters/second. Could you outrun a pyroclastic flow on foot, on a bicycle, or in a car?

Locate the numbers \(-4.5\), \(-0.75\), \(\dfrac{1}{2}\), \(\sqrt{2}\), \(\pi\), and \(4.1\) on the real number line.

Suppose a sample space is a deck of \(52\) playing cards. Let set \(A=\{ \text{Aces} \}\) and \(B=\{ \text{Kings} \}\) and use a Venn diagram to show that \(A\) and \(B\) are mutually exclusive.

Use a Venn diagram to show the intersection of the set \(A=\{\text{Aces}\}\) and \(B=\{\text{Spades}\}\) from the sample space of a deck of playing cards.

The Venn diagram below can be used to model the relationships among three categories of fruit.

Set A:Fruits that are red (e.g., apples, red peppers,...)

Set B: Fruits that are sweet (e.g., oranges, strawberries...)

Set C: Fruits that are roundish (e.g., grapes, apples, ...)

Overlapping regions (sets) show fruits that belong to multiple categories. For example, a red, sweet, and round fruit like a red apple would be in the intersection of all three sets, which is Region (or set) G.

a. What category of fruit would you find in region E? Give an example.

b. A fruit in Region H would be sweet and roundish, but not red. True or false?

c. If we consider bananas to be sweet, in what region or regions would you find them?

d. A fruit in Region E would be red and roundish, but not sweet. True or false?

e. Name a fruit in Region G.

Suppose you want to classify the numbers from \(1\) to \(20\) into three categories: Even numbers, Prime numbers, and Multiples of \(5\). Here are the three sets of numbers:

Set A Even numbers from 1 to 20:\(\{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}\)

Set B Prime numbers from 1 to 20: \(\{2, 3, 5, 7, 11, 13, 17, 19\}\)

Set C Multiples of 5:\(\{5, 10, 15, 20\}\)

Create a Venn diagram to show the relationships between these sets.