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”.


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:


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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