Math / Algebra

QuestionIf you practice your magic tricks often then you won't mess them up on stage, and if you don't practice your magic tricks often then you will mess them up on stage. A. Write the statement in symbolic form. B. Construct a truth table for the symbolic statement in part A. C. Use the truth table to indicate one set of conditions that make the compound statement false, or state that no such conditions exist. A. Write the statement in symbolic form.
Let $p=$ "you practice your magic tricks often" and $q=$ "you will mess them up on stage." A. $(p \rightarrow \sim q) \wedge(\sim p \rightarrow q)$ B. $(p \rightarrow \sim q) \vee(\sim p \rightarrow q)$ C. $(p \leftrightarrow \sim q) \vee(\sim p \rightarrow q)$ D. $(p \rightarrow \sim q) \wedge(\sim p \leftrightarrow q)$

Studdy Solution

STEP 1

What is this asking? We're checking if practicing magic tricks prevents messing up on stage, translating this into symbols, making a truth table, and finding when the statement is false. Watch out! Don't mix up the arrow symbols! $\rightarrow$ means "if...then," $\wedge$ means "and," and $\leftrightarrow$ means "if and only if."

STEP 2

1. Symbolize the statement
2. Build the truth table
3. Find the false conditions

STEP 3

Let $p$ be "You practice often," and let $q$ be "You mess up on stage." So, $\sim q$ means "You don't mess up on stage."

STEP 4

"If you practice often, then you don't mess up" becomes $p \rightarrow \sim q$ .

STEP 5

"If you don't practice often, then you mess up" becomes $\sim p \rightarrow q$ .

STEP 6

Since both parts must be true, we use "and": $(p \rightarrow \sim q) \wedge (\sim p \rightarrow q)$ .
This matches answer choice A!

STEP 7

We need columns for $p$ , $q$ , $\sim p$ , $\sim q$ , $p \rightarrow \sim q$ , $\sim p \rightarrow q$ , and finally, the whole statement $(p \rightarrow \sim q) \wedge (\sim p \rightarrow q)$ .

STEP 8

These are our base truth values: True/True, True/False, False/True, False/False.

STEP 9

Just flip the truth values of p and q, respectively.

STEP 10

Remember, "if...then" is only false when the first part is true and the second part is false.

STEP 11

Same rule as before: only false when the first part ( $\sim p$ ) is true and the second part ( $q$ ) is false.

STEP 12

Now, we look at the "and" ( $\wedge$ ) of the two previous columns. "And" is only true when both parts are true.

STEP 13

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Studdy Solution

STEP 1

STEP 2

1. Symbolize the statement
2. Build the truth table
3. Find the false conditions

STEP 3

Let $p$ be "You practice often," and let $q$ be "You mess up on stage." So, $\sim q$ means "You don't mess up on stage."

STEP 4

"If you practice often, then you don't mess up" becomes $p \rightarrow \sim q$ .

STEP 5

"If you don't practice often, then you mess up" becomes $\sim p \rightarrow q$ .

STEP 6

Since both parts must be true, we use "and": $(p \rightarrow \sim q) \wedge (\sim p \rightarrow q)$ .
This matches answer choice A!

STEP 7

We need columns for $p$ , $q$ , $\sim p$ , $\sim q$ , $p \rightarrow \sim q$ , $\sim p \rightarrow q$ , and finally, the whole statement $(p \rightarrow \sim q) \wedge (\sim p \rightarrow q)$ .

STEP 8

These are our base truth values: True/True, True/False, False/True, False/False.

STEP 9

Just flip the truth values of p and q, respectively.

STEP 10

Remember, "if...then" is only false when the first part is true and the second part is false.

STEP 11

Same rule as before: only false when the first part ( $\sim p$ ) is true and the second part ( $q$ ) is false.

STEP 12

Now, we look at the "and" ( $\wedge$ ) of the two previous columns. "And" is only true when both parts are true.

STEP 13

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Studdy Solution

STEP 1

STEP 2

1. Symbolize the statement2. Build the truth table3. Find the false conditions

STEP 3

Let ppp be "You practice often," and let qqq be "You mess up on stage." So, ∼q\sim q∼q means "You *don't* mess up on stage."

STEP 4

"If you practice often, then you *don't* mess up" becomes p→∼q p \rightarrow \sim q p→∼q.

STEP 5

"If you *don't* practice often, then you mess up" becomes ∼p→q \sim p \rightarrow q ∼p→q.

STEP 6

Since both parts must be true, we use "and": (p→∼q)∧(∼p→q) (p \rightarrow \sim q) \wedge (\sim p \rightarrow q) (p→∼q)∧(∼p→q).This matches answer choice **A**!

STEP 7

We need columns for ppp, qqq, ∼p\sim p∼p, ∼q\sim q∼q, p→∼qp \rightarrow \sim qp→∼q, ∼p→q\sim p \rightarrow q∼p→q, and finally, the whole statement (p→∼q)∧(∼p→q) (p \rightarrow \sim q) \wedge (\sim p \rightarrow q) (p→∼q)∧(∼p→q).

STEP 8

These are our **base truth values**: True/True, True/False, False/True, False/False.

STEP 9

Just flip the truth values of *p* and *q*, respectively.

STEP 10

Remember, "if...then" is only false when the first part is true and the second part is false.

STEP 11

Same rule as before: only false when the first part (∼p\sim p∼p) is true and the second part (qqq) is false.

STEP 12

Now, we look at the "and" (∧ \wedge ∧) of the two previous columns. "And" is only true when *both* parts are true.

STEP 13

[ \begin{array}{|c|c|c|c|c|c|c|} \hline

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1. Symbolize the statement
2. Build the truth table
3. Find the false conditions

Let $p$ be "You practice often," and let $q$ be "You mess up on stage." So, $\sim q$ means "You don't mess up on stage."

"If you practice often, then you don't mess up" becomes $p \rightarrow \sim q$ .

"If you don't practice often, then you mess up" becomes $\sim p \rightarrow q$ .

Since both parts must be true, we use "and": $(p \rightarrow \sim q) \wedge (\sim p \rightarrow q)$ .
This matches answer choice A!

We need columns for $p$ , $q$ , $\sim p$ , $\sim q$ , $p \rightarrow \sim q$ , $\sim p \rightarrow q$ , and finally, the whole statement $(p \rightarrow \sim q) \wedge (\sim p \rightarrow q)$ .

These are our base truth values: True/True, True/False, False/True, False/False.

Just flip the truth values of p and q, respectively.

Same rule as before: only false when the first part ( $\sim p$ ) is true and the second part ( $q$ ) is false.

Now, we look at the "and" ( $\wedge$ ) of the two previous columns. "And" is only true when both parts are true.