MHB How Do Truth Tables and Venn Diagrams Verify Logical Equivalence?

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SUMMARY

This discussion focuses on using truth tables and Venn diagrams to verify logical equivalence in propositional logic. Specifically, it demonstrates that the expression p v (q ^ r) is equivalent to (p v q) ^ (p v r) through a detailed truth table analysis. The discussion also clarifies that this expression is not equivalent to (p v q) ^ r. Participants provided a structured truth table to illustrate these relationships, ensuring clarity in the logical operations involved.

PREREQUISITES
  • Understanding of propositional logic
  • Familiarity with truth tables
  • Knowledge of Venn diagrams
  • Basic skills in logical equivalence
NEXT STEPS
  • Study the construction of truth tables for complex logical expressions
  • Explore Venn diagram representations of logical statements
  • Learn about logical equivalence proofs in propositional logic
  • Investigate the implications of De Morgan's laws in logical expressions
USEFUL FOR

This discussion is beneficial for students of mathematics, educators teaching logic, and anyone interested in enhancing their understanding of logical equivalence and its verification methods.

barbara
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How would I use a truth table to show that the statement p v (q ^ r) is equivalent to (p v q) ^ (p v r) or design a venn diagram for this. and show that this statement is not equivalent to (p v q) ^ r.
 
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A truth-table might begin like this:

$\begin{array}{cccccccc}

P&Q& R&Q\wedge R&P\vee(Q\wedge R)& P\vee Q& P\vee R& (P\vee Q)\wedge(P\vee R)\\
\ast&\ast&\ast&\ast&\ast&\ast&\ast&\ast\\
T&T&T&T&T&T&T&-\\
T&T&F&F&T&T&T&-\\
T&F&T&F&T&T&T&-\\
T&F&F&F&T&T&T&-\\
F&T&T&T&T&T&T&-\\
F&T&F&F&F&T&F&-\\
F&F&T&F&F&F&T&-\\
F&F&F&F&F&F&F&-\\

\end{array}$

Your goal, then, is to fill out column 8, and verify it matches column 5...
 
If there are an infinite number of natural numbers, and an infinite number of fractions in between any two natural numbers, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and... then that must mean that there are not only infinite infinities, but an infinite number of those infinities. and an infinite number of those...

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