“Rules of inferences” is an important topic in Discrete Mathematics. We can use a rule (s) of inferences to prove if an argument is valid or not. We can also use rules of inference to produce valid arguments.

In mathematics, an argument is a sequence of statements. We have a valid argument if and only if is impossible for all the premises to be true, and the conclusion to be false.

We can decide whether an argument is valid or not by using rules of inference.

Let’s see the solution to an exercise.

## Table of Contents

- a) Alice is a mathematics major. Therefore, Alice is either a mathematics major or a computer science major.
- b) Jerry is a mathematics major and a computer science major. Therefore, Jerry is a mathematics major.
- c) If it is rainy, then the pool will be closed. It is rainy. Therefore, the pool is closed.
- d) If it snows today, the university will close. The university is not closed today. Therefore, it did not snow today.
- e) If I go swimming, then I will stay in the sun too long. If I stay in the sun too long, then I will sunburn. Therefore, if I go swimming, then I will sunburn.

What rule of inference is used in each of these arguments?

**a) Alice is a mathematics major. Therefore, Alice is either a mathematics major or a computer science major**.

Let p=”Alice is a mathematics major” and q=”Alice is a computer science major”

Rewriting the argument, we have:

```
p
------
∴p∨q
```

**Answer: Addition. **

We can also affirm that this is a valid argument because of the ** rule of Addition**.

**b) Jerry is a mathematics major and a computer science major. Therefore, Jerry is a mathematics major.**

Following a similar reasoning to the previous exercise:

p = “Jerry is a mathematics major”, q = “Jerry is a computer science major”

```
p^q
------
∴p
```

**Answer:** ** Simplification**.

**c) If it is rainy, then the pool will be closed. It is rainy. Therefore, the pool is closed.**

p=”it is rainy”, q=”the pool will be closed”

```
p
p->q
------
∴q
```

**Answer: Modus Ponens.**

**d) If it snows today, the university will close. The university is not closed today. Therefore, it did not snow today.**

p=”it snows today”, q=”the university will close”

```
p->q
¬q
-------
∴¬q
```

**Answer**: ** Modus Tollens**.

**e) If I go swimming, then I will stay in the sun too long. If I stay in the sun too long, then I will sunburn. Therefore, if I go swimming, then I will sunburn.**

p=”I go swimming”, q=”I will stay in the sun too long”, r=”I will sunburn”

```
p->q
q->r
--------
∴q->r
```

**Answer**: __Hypothetical syllogism__

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