Proving a tautology using truth table

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SUMMARY

The discussion centers on proving the statement ((p_r)^(q_r))<-->((p^q)_r) as a tautology using a truth table. The key conclusion is that the left-hand side (LHS) of the biconditional must equal the right-hand side (RHS) to establish it as a tautology. Participants clarified the symbols used, identifying ^ as conjunction (AND) and _ as disjunction (inclusive OR), which may vary by educational context. The use of truth tables is emphasized as a fundamental method for evaluating logical statements.

PREREQUISITES
  • Understanding of logical operators: conjunction (AND) and disjunction (inclusive OR).
  • Familiarity with truth tables and their construction.
  • Basic knowledge of biconditional statements in propositional logic.
  • Awareness of symbolic notation variations in logic across different educational contexts.
NEXT STEPS
  • Study the construction and interpretation of truth tables in propositional logic.
  • Learn about biconditional statements and their properties in formal logic.
  • Explore different logical symbols and their meanings in various logic courses.
  • Investigate advanced topics such as tautologies, contradictions, and contingencies in logical expressions.
USEFUL FOR

Students of logic, educators teaching propositional logic, and anyone interested in understanding logical proofs and truth tables.

MarcL
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Homework Statement


((p_r)^(q_r))<-->((p^q)_r) Prove whether the following statement is a tautology , contigency or contradiction using a truth table.

Homework Equations

The Attempt at a Solution


I did the truth table, but this whole thing is one statement no? What do I compare? the first half of the bi conditional statement to the second? or the whole statement vs. the two halves? I don't know which to compare to decide whether or not it is a contingency.EDIT: figured out the tautology is if L.H.S of <--> is equal to RHS. Sorry for the post!
 
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Just a question for you, what is ^ and _? In the logic courses I took, we used different symbols and I'm aware different schools/fields use different symbols so just wanted to know what yours are. Thanks :)

Btw I'm guessing ^ is conjunction and _ is disjunction?
 
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Well I used ^ for conjunction ( so AND) and _ for inclusive or because I can't the the opposite of ^ :) ( it was my own guess)
 
MarcL said:
Well I used ^ for conjunction ( so AND) and _ for inclusive or because I can't the the opposite of ^ :) ( it was my own guess)

Ah, alright :D If you click the sigma Σ button on the toolbar, there are many symbols to choose from including ∧ ∨ ⊃ ⋅ ∴
 

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