# Lab 10: Introduction to Logic
## Problem 1: Back and Forth
**a. Atomic Propositions:**
- p1 = It rained this morning
- p2 = It will be sunny this afternoon

**Formal Proposition:**
$p1 \wedge p2$

**b. Atomic Propositions:**
- p1 = I am forced to stay in-doors this week
- p2 = I will play Genshin Impact
- p3 = I will have fun
- p4 = I will become poor

**Formal Proposition:**
$p1 \rightarrow p2 \wedge p3 \wedge p4$

**c. Atomic Propositions:**
- p1 = I eat taco bell
- p2 = I eat mcdonalds
- p3 = I will become swoll
- p4 = I will survive the night

**Formal Proposition:**
$p1 \vee p2 \rightarrow p3 \vee \neg p4$

**d. Atomic Propositions:**
- p1 = Jan works at the company
- p2 = Bob works at the company
- p3 = Jan works more than Bob

**Formal Proposition:**
$p1 \wedge p2 \rightarrow p3$

### **Natural Language Propositions**
**a.** I love cats and cats barfs.
**b.** If I love cats, then I do not own dogs.
**c.** Either I love cats or, if cat barfs then I own dog.
**d.** If my cat barfs then I'm sending away my cat or I do not love my cat.
**e.** If I love dogs, then I own dogs or I'm sending away cats.

## Problem 2: Thinking About Proof
**Atomic Propositions**
>a. In order to prove an atomic proposition, we would have to have knowledge of the described variables.

>b. For example, in order to prove the previous atomic proposition of "I eat taco bell," we would require knowledge of your eating habits in order to prove the proposition. Without this not-provided information, we cannot but assume that the proposition is true.

**Compound Propositions**
>a. In order to prove a compound proposition, we must split it into its most basic parts: atomic propositions. Then we can apply logic to further redefine and understand each proposition until we agree that they are true. Because our reasoning based on these propositions is accurate, our compound proposition must also be true.

>b. For example, in order to prove the compound proposition, "If Biden is president, and Trump is not leading a revolution, then the wall is coming down," we must split it into atomic propositions.
p1. Biden is president
p2. Trump is not leading a revolution
p3. The wall is coming down
Since Biden is, in fact, president, and Trump is not (yet) leading a revolution, then it follows logically that the wall is coming down.