Math / Algebra

QuestionTranslate the argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form.
You drew a three, or you have two pairs. Today you do not have two pairs. $\therefore$ Today you drew a three. (i) Click the icon to view tables of standard valid and invalid forms of arguments.
Let p represent "Today you drew a three.," and let q represent "Today you have two pairs." Select the correct choice below and fill in the answer box with the symbolic form of the argument. (Type the terms of your expression in the same order as they appear in the original expression.) A. The argument is valid. In symbolic form the argument is $\square$ B. The argument is invalid. In symbolic form the argument is $\square$

Studdy Solution

STEP 1

What is this asking? We need to convert a word problem into symbols, and then figure out if the conclusion is guaranteed based on the given facts. Watch out! Don't mix up "or" with "and".
Also, be careful with negation – "not having two pairs" is different from "having two pairs"!

STEP 2

1. Symbolize the statements
2. Analyze the argument

STEP 3

Let's define what our letters mean. $p$ means "Today you drew a three," and $q$ means "Today you have two pairs." So exciting!

STEP 4

The first sentence, "You drew a three, or you have two pairs," translates to $p \lor q$ .
The symbol $\lor$ means "or".

STEP 5

The second sentence, "Today you do not have two pairs," translates to $\neg q$ .
The symbol $\neg$ means "not".

STEP 6

The conclusion, "Today you drew a three," is simply $p$ .

STEP 7

Putting it all together, our argument looks like this: $\begin{aligned} & p \lor q \\ & \neg q \\ & \therefore p \end{aligned}$ This structure tells us: if the first two statements are true, then the conclusion must also be true.
Let's see if that's the case!

STEP 8

Let's think this through.
We're told that either $p$ is true (you drew a three) or $q$ is true (you have two pairs), or both could be true.
That's what "or" means.

STEP 9

We're also told that $q$ is not true (you don't have two pairs).

STEP 10

So, if $q$ is false, and we know that at least one of $p$ or $q$ must be true, then $p$ has to be true!
It's the only possibility left!

STEP 11

This type of argument is called "disjunctive syllogism," and it's a valid argument form.
This means that if the premises (the first two statements) are true, then the conclusion (the last statement) absolutely must be true!

STEP 12

The argument is valid.
In symbolic form, the argument is: $\begin{aligned} & p \lor q \\ & \neg q \\ \therefore \ & p \end{aligned}$ We choose answer A.

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Studdy Solution

STEP 1

What is this asking? We need to convert a word problem into symbols, and then figure out if the conclusion is guaranteed based on the given facts. Watch out! Don't mix up "or" with "and".
Also, be careful with negation – "not having two pairs" is different from "having two pairs"!

STEP 2

1. Symbolize the statements
2. Analyze the argument

STEP 3

Let's define what our letters mean. $p$ means "Today you drew a three," and $q$ means "Today you have two pairs." So exciting!

STEP 4

The first sentence, "You drew a three, or you have two pairs," translates to $p \lor q$ .
The symbol $\lor$ means "or".

STEP 5

The second sentence, "Today you do not have two pairs," translates to $\neg q$ .
The symbol $\neg$ means "not".

STEP 6

The conclusion, "Today you drew a three," is simply $p$ .

STEP 7

Putting it all together, our argument looks like this: $\begin{aligned} & p \lor q \\ & \neg q \\ & \therefore p \end{aligned}$ This structure tells us: if the first two statements are true, then the conclusion must also be true.
Let's see if that's the case!

STEP 8

Let's think this through.
We're told that either $p$ is true (you drew a three) or $q$ is true (you have two pairs), or both could be true.
That's what "or" means.

STEP 9

We're also told that $q$ is not true (you don't have two pairs).

STEP 10

So, if $q$ is false, and we know that at least one of $p$ or $q$ must be true, then $p$ has to be true!
It's the only possibility left!

STEP 11

This type of argument is called "disjunctive syllogism," and it's a valid argument form.
This means that if the premises (the first two statements) are true, then the conclusion (the last statement) absolutely must be true!

STEP 12

The argument is valid.
In symbolic form, the argument is: $\begin{aligned} & p \lor q \\ & \neg q \\ \therefore \ & p \end{aligned}$ We choose answer A.

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Studdy Solution

STEP 1

What is this asking? We need to convert a word problem into symbols, and then figure out if the conclusion is guaranteed based on the given facts. Watch out! Don't mix up "or" with "and".Also, be careful with negation – "not having two pairs" is different from "having two pairs"!

STEP 2

1. Symbolize the statements2. Analyze the argument

STEP 3

Let's **define** what our letters mean. ppp means "Today you drew a three," and qqq means "Today you have two pairs." So exciting!

STEP 4

The first sentence, "You drew a three, or you have two pairs," translates to p∨qp \lor qp∨q.The symbol ∨\lor∨ means "or".

STEP 5

The second sentence, "Today you do not have two pairs," translates to ¬q\neg q¬q.The symbol ¬\neg¬ means "not".

STEP 6

The conclusion, "Today you drew a three," is simply ppp.

STEP 7

Putting it all together, our argument looks like this: p∨q¬q∴p\begin{aligned} & p \lor q \\ & \neg q \\ & \therefore p \end{aligned}​p∨q¬q∴p​ This structure tells us: *if* the first two statements are true, *then* the conclusion *must* also be true.Let's see if that's the case!

STEP 8

Let's think this through.We're told that *either* ppp is true (you drew a three) *or* qqq is true (you have two pairs), or *both* could be true.That's what "or" means.

STEP 9

We're *also* told that qqq is *not* true (you *don't* have two pairs).

STEP 10

So, if qqq is false, and we know that at least one of ppp or qqq *must* be true, then ppp *has* to be true!It's the only possibility left!

STEP 11

This type of argument is called "disjunctive syllogism," and it's a **valid** argument form.This means that *if* the premises (the first two statements) are true, then the conclusion (the last statement) *absolutely must* be true!

STEP 12

The argument is **valid**.In symbolic form, the argument is: p∨q¬q∴ p\begin{aligned} & p \lor q \\ & \neg q \\ \therefore \ & p \end{aligned}∴ ​p∨q¬qp​ We choose answer A.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We need to convert a word problem into symbols, and then figure out if the conclusion is guaranteed based on the given facts. Watch out! Don't mix up "or" with "and".
Also, be careful with negation – "not having two pairs" is different from "having two pairs"!

1. Symbolize the statements
2. Analyze the argument

Let's define what our letters mean. $p$ means "Today you drew a three," and $q$ means "Today you have two pairs." So exciting!

The first sentence, "You drew a three, or you have two pairs," translates to $p \lor q$ .
The symbol $\lor$ means "or".

The second sentence, "Today you do not have two pairs," translates to $\neg q$ .
The symbol $\neg$ means "not".

The conclusion, "Today you drew a three," is simply $p$ .

Putting it all together, our argument looks like this: $\begin{aligned} & p \lor q \\ & \neg q \\ & \therefore p \end{aligned}$ This structure tells us: if the first two statements are true, then the conclusion must also be true.
Let's see if that's the case!

Let's think this through.
We're told that either $p$ is true (you drew a three) or $q$ is true (you have two pairs), or both could be true.
That's what "or" means.

We're also told that $q$ is not true (you don't have two pairs).

So, if $q$ is false, and we know that at least one of $p$ or $q$ must be true, then $p$ has to be true!
It's the only possibility left!

This type of argument is called "disjunctive syllogism," and it's a valid argument form.
This means that if the premises (the first two statements) are true, then the conclusion (the last statement) absolutely must be true!

The argument is valid.
In symbolic form, the argument is: $\begin{aligned} & p \lor q \\ & \neg q \\ \therefore \ & p \end{aligned}$ We choose answer A.