If…Then Statements: Problems from Class!

1. Take the abstract mathematical statement represented here with variables: P ⇒ Q. 
Under what conditions is this statement taken to be true? False?

 To find the answer, go here! The only other information you need is that P represents the hypothesis and Q represents the conclusion.

2. Five more mathematical statements to define! (Define P, Q, and give relationship based on what the speaker means)

A. If it’s sunny in the morning, I cycle to work. 

P: If it’s sunny in the morning

Q: Then I cycle to work

Relationship: The sunny morning is causing this person to cycle to work

B: If you eat your peas, then you get ice cream!

P: If you eat your peas

Q: Then you get ice cream

Relationship: The fact that you eat your peas (what a noble venture, by the way), is causing you to get the (awesome) reward of ice cream

C: When Gotham is ashes, you have my permission to die.

First of all, how very morbid.

P: If Gotham at some point becomes ashes

Q: Then you will have Bane’s permission to die (Yay….!)

Relationship: Gotham going up in flames is giving cause to your death.

D: If you elect me, your taxes will go down!

P: If you elect me

Q: Then taxes will decrease (ahahahaha)

Relationship: Electing someone will be the causality for taxes decreasing

E: If a number has only itself and one as factors, we the call the number prime.

P: If a number has only itself and one as factors

Q: Then that number is considered prime

Relationship: The conditions of a prime number being prime gives cause to calling that number prime.

3. Find the truth set for “p is an even prime”.

{2}

4. Find the truth set for “The function f(x) = x2 + 2x + a has two roots”

To find roots for an equation, we can use the quadratic formula! The formula, for the majority of us who don’t remember algebra two, is:3ea647783b5121989cd87ca3bb558916

Luckily, we already know what a and b are, so we can plug them into the equation, simplify, and get the equation:

-1±√1+c

Roots can be real or unreal, so having c be a negative number is not an issue. The only issue we run into is when c = -1. At that point, the equation only has one root. So, the truth set is:

{all real numbers where c ≠ -1.}

5. Find the truth set for “A ⊆ {1,2,3,4,5} and |A| ≤ 2”

First of all, the notation |A| simply means that you are counting up the number of elements in a set. So the truth set must contain sets that have two elements each or less.

{{  },{1},{2},{3},{4},{5},{1,2},{1,3},{1,4},{1,5},{2,3},{2,4},{2,5},{3,4},{3,5},{4,5}}

This set includes the empty set, sets with only one number from the A subset, and sets with 2 numbers from the A set.

6. Find the truth set for  “A,B⊆ {1,2,3,4,5} with A⊆B and |B| – |A| = 4”

To unpack this meaning, first we realize that A and B are both sets made up of the numbers 1, 2, 3, 4 and/or 5. The second part of this open sentence is saying that A is a subset of B, which means that all elements of A are in B as well. Finally, the last part means that the set B must either have 5 or 4 elements in it, with 1 or 0 elements in A, respectively. So this truth set will be pretty large…

{ (A = { }, B={1,2,3,4}) – and all other possible combinations of 4 elements of A & B, where A is the empty set.

(A= {1}, B={1,2,3,4,5}) – and all other possible combinations of A equaling different elements from the subset.}

And now for the most important part:

BONUS: “Strong men also cry.” What movie is this from??

The Big Lebowski. Obviously.

 

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