Are (A then B) then C and (A and B) then C equivalent?

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SUMMARY

The discussion focuses on proving the equivalence of the logical expressions "(A then B) then C" and "(A and B) then C" using a truth table. It is established that there are 23 possible combinations of truth values for A, B, and C, but only 8 unique combinations are relevant for this analysis. The expressions are equivalent if they yield the same truth values across all combinations, which can be verified by evaluating each scenario in the truth table format.

PREREQUISITES
  • Understanding of basic logical operators: "then" (implication) and "and".
  • Familiarity with truth tables and their construction.
  • Knowledge of logical expressions and their evaluation.
  • Basic skills in propositional logic.
NEXT STEPS
  • Learn how to construct and analyze truth tables for complex logical expressions.
  • Study the principles of propositional logic, focusing on implications and conjunctions.
  • Explore logical equivalences and their proofs in mathematical logic.
  • Investigate the use of truth tables in computer science, particularly in circuit design.
USEFUL FOR

This discussion is beneficial for students studying precalculus, educators teaching logic concepts, and anyone interested in understanding logical equivalences in mathematical reasoning.

Gamecockgirl
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I need help proving that (A then B) then C and (A and B) then C are equivalent. Can anyone help?
 
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This doesn't meet criterai for posting in Philosophy. If this is homework, please post in the appropriate Homework forum
 
I am moving this to the "precalculus homework" forum.

Gamecock Girl, I think the simplest way to do this is to use a "truth table". There are 23 different ways A, B, and C can be "True" or "False". The two statements are equivalent if they are both True or both False in each of those.

Writing "T" for True, "F" for False and "ABC" in that order, the 8 ways are:
TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF. Assuming you know what "if (if A then B) then C" and "if (A and B) then C" mean, you can decide whether they are True of False in each of those 8 situations.
 

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