Math Problem Statement

Construct a truth table for the statement.

r (r ~r)

Solution

To construct a truth table for the statement r(r¬r)r \land (r \lor \neg r), we need to consider all possible truth values for the variable rr and then evaluate each part of the expression step by step.

Here's the breakdown:

  1. rr - the initial variable.
  2. ¬r\neg r - the negation of rr.
  3. r¬rr \lor \neg r - the disjunction (logical OR) of rr and ¬r\neg r.
  4. r(r¬r)r \land (r \lor \neg r) - the conjunction (logical AND) of rr and the result from step 3.

Now, let's fill in the truth table:

rr¬r\neg rr¬rr \lor \neg rr(r¬r)r \land (r \lor \neg r)
TFTT
FTTF

Explanation:

  • Column 1 (rr): Lists the possible truth values of rr (True and False).
  • Column 2 (¬r\neg r): Lists the negation of rr.
  • Column 3 (r¬rr \lor \neg r): This is always True because rr or ¬r\neg r covers all possibilities (one of them will always be true).
  • Column 4 (r(r¬r)r \land (r \lor \neg r)): The result depends on rr. If rr is True, the result is True; if rr is False, the result is False.

Would you like to go into more detail or have any questions?


Here are five related questions you might explore next:

  1. What is the truth table for ¬(rs)\neg(r \land s)?
  2. How does De Morgan's Law apply to logical expressions?
  3. Can you create a truth table for rs¬tr \land s \lor \neg t?
  4. What is the difference between a tautology and a contradiction?
  5. How do truth tables help in understanding logical equivalences?

Tip: Practice constructing truth tables for different logical expressions to get a better grasp of how each logical operator works.

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Math Problem Analysis

Mathematical Concepts

Logic
Truth Tables

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12