Questions: Use DeMorgan's Laws to write an equivalent statement in symbolic form. Select the correct choice. -[(p ∧ q) ∨ r] (p ∨ q) ∧ -r (p ∧ q) ∨ r (p ∨ ∼ q) ∧ ∼ r (-p ∧ ∼ q) ∨ r (∼ p ∨ q) ∧ ∼ r (∼ p ∨ ∼ q) ∧ ∼ r Submit Question

Use DeMorgan's Laws to write an equivalent statement in symbolic form. Select the correct choice.

-[(p ∧ q) ∨ r]

(p ∨ q) ∧ -r

(p ∧ q) ∨ r

(p ∨ ∼ q) ∧ ∼ r

(-p ∧ ∼ q) ∨ r

(∼ p ∨ q) ∧ ∼ r

(∼ p ∨ ∼ q) ∧ ∼ r

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Transcript text: Use DeMorgan's Laws to write an equivalent statement in symbolic form. Select the correct choice. \[ -[(p \wedge q) \vee r] \] $(p \vee q) \wedge-r$ $(p \wedge q) \vee r$ $(p \vee \sim q) \wedge \sim r$ $(-p \wedge \sim q) \vee r$ $(\sim p \vee q) \wedge \sim r$ $(\sim p \vee \sim q) \wedge \sim r$ Submit Question

Solution

Solution Steps

To apply DeMorgan's Laws, we need to distribute the negation across the logical operators inside the expression. DeMorgan's Laws state that the negation of a conjunction is the disjunction of the negations, and vice versa. Therefore, we will apply these laws to the given expression to find an equivalent statement.

Step 1: Original Expression

The original expression we need to analyze is given by: \[ -\left[(p \wedge q) \vee r\right] \]

Step 2: Apply DeMorgan's Laws

Using DeMorgan's Laws, we can rewrite the negation of the disjunction: \[ -\left[(p \wedge q) \vee r\right] = -r \wedge -(p \wedge q) \]

Step 3: Further Simplification

Next, we apply DeMorgan's Laws again to the term $-(p \wedge q)$: \[ -(p \wedge q) = -p \vee -q \] Thus, we can rewrite the expression as: \[ -\left[(p \wedge q) \vee r\right] = -r \wedge (-p \vee -q) \] This can be expressed as: \[ (-r) \wedge (\sim p \vee \sim q) \]