Math  /  Discrete

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 contributed money, or we had enough. Last week we had enough. \therefore Last week you did not contribute money. Click the icon to view tables of standard valid and invalid forms of arguments.
Let p represent "Last week you contributed money.," and let q represent "Last week we had enough." 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 rewrite the word problem using symbols, then figure out if the conclusion follows logically from the given facts. Watch out! Don't mix up the order of the statements when converting to symbols, and be careful with negation!

STEP 2

1. Define the propositions
2. Symbolize the argument
3. Analyze the argument

STEP 3

Let's **define** what our letters mean! pp represents "You contributed money," and qq represents "We had enough." Super clear, right?

STEP 4

The first statement, "You contributed money, or we had enough," translates to pqp \lor q.
The little \lor symbol means "or."

STEP 5

The second statement, "Last week we had enough," is simply qq.

STEP 6

The conclusion, "Last week you did not contribute money," is written as ¬p\neg p.
That little ¬\neg means "not."

STEP 7

So, the entire argument in symbolic form is: pqp \lor q, qq \therefore ¬p\neg p.
The \therefore symbol means "therefore".

STEP 8

Let's **test** this out with a simple example.
Suppose pp is *false* (you didn't contribute money) and qq is *true* (we had enough).

STEP 9

Does the first statement, pqp \lor q, hold true?
Yes! Because even though pp is false, qq is true, and "false or true" is true.
Remember, "or" only needs *one* part to be true!

STEP 10

The second statement, qq, is true (we assumed that from the start).

STEP 11

Now, the conclusion, ¬p\neg p, is "not pp." Since pp is false, ¬p\neg p (not pp) is true!

STEP 12

So, we have a situation where both premises (the first two statements) are true, and the conclusion is also true.
But is the argument *always* valid?

STEP 13

Let's consider another case.
What if *both* pp and qq are true?
That is, you *did* contribute money, and we *still* had enough.

STEP 14

The first statement, pqp \lor q, is true (true or true is true).
The second statement, qq, is also true.

STEP 15

But the conclusion, ¬p\neg p (not pp), is now *false*!
We assumed pp was true, so "not pp" is false.
Uh oh!

STEP 16

We found a case where the premises are true, but the conclusion is false.
This means the argument is **invalid**!

STEP 17

The argument is invalid.
In symbolic form, the argument is pqp \lor q, qq \therefore ¬p\neg p.

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