Please type your answer. Thank you Certainly! Here is a transcription of the text suitable for an educational website:---**Problem:**Show the following equivalence using both truth tables and the laws of logic. In your logical solution, justify each of your steps by stating which law you are using.\[ P \leftrightarrow Q \text{ is equivalent to } \neg P \leftrightarrow \neg Q. \]### Explanation:In this problem, you are asked to prove that the statement “P is equivalent to Q” is logically equivalent to “not P is equivalent to not Q.” This can be done using truth tables and the laws of logic.1. **Using Truth Tables:** - Construct a truth table for both the expression \( P \leftrightarrow Q \) and \( \neg P \leftrightarrow \neg Q \). - Compare the columns for logical equivalence.2. **Using Laws of Logic:** - Apply logical laws, such as De Morgan's Theorems or the Law of Implication, to demonstrate the steps of the equivalence.---This task encourages understanding of logical equivalence through both analytical and procedural methods.

Name: Please type your answer. Thank you Certainly! Here is a transcription
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: Please type your answer. Thank you Certainly! Here is a transcription of the text suitable for an educational website:---**Problem:**Show the following equivalence using both truth tables and the laws of logic. In your logical solution, justify each

We have to prove that P↔Q is equivalent to ¬P↔¬Q. Proof using truth table: We know that if a proposition A is true then its negation ¬A is false and if A is false then its negation ¬A is true. Also, a double implication A↔B is true if and only if both A and B have the same truth values. Otherwise it is false. There are 4 different combinations of truth values for variables P and Q. Construct the truth table for P↔Q and ¬P↔¬Q corresponding to those 4 combinations as follows. P Q ¬P ¬Q P↔Q ¬P↔¬Q T T F F T T T F F T F F F T T F F F F F T T T T From the last 2 columns of the above truth table, it can be found that P↔Q and ¬P↔¬Q have identical truth values for all possible combinations of truth values of P and Q. Hence, P↔Q is equivalent to ¬P↔¬Q. Proof using laws of logic: We have the following laws of logic. Material equivalence: A↔B≡A→B∧B→A Law of contrapositive: A→B≡¬B→¬A We use the above laws to prove that, P↔Q is equivalent to ¬P↔¬Q. P↔Q≡P→Q∧Q→P Material equivalence with A~P and B~Q≡¬Q→¬P∧¬P→¬Q Law of contrapositive P→Q≡¬Q→¬P and Q→P≡¬P→¬Q≡¬P→¬Q∧¬Q→¬P Commutativity≡¬P↔¬Q Material equivalence with A~¬P and B~¬Q Hence, P↔Q is equivalent to ¬P↔¬Q.