Logical Equivalence of x <=> y and (x-->y) ^ ((~x)-->(~y))

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SUMMARY

The logical equivalence of x <=> y is established as (x-->y) ^ ((~x)-->(~y)). This equivalence can be proven using a truth table, which outlines the conditions under which both expressions yield the same truth values. The discussion emphasizes the importance of understanding the contrapositive and suggests that those unfamiliar with truth tables may struggle with mathematical concepts. Specific references to proof techniques and the need for clarity in mathematical reasoning are highlighted.

PREREQUISITES
  • Understanding of logical operators: implication (-->), negation (~), and biconditional (<=>
  • Familiarity with truth tables for evaluating logical expressions
  • Basic knowledge of mathematical proofs and equivalences
  • Concept of contrapositives in logic
NEXT STEPS
  • Learn how to construct and analyze truth tables for logical expressions
  • Study the concept of contrapositives and their role in logical reasoning
  • Explore formal proof techniques in propositional logic
  • Review logical equivalences and their applications in computer science
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Students of mathematics, computer science majors, and anyone interested in understanding logical reasoning and proof techniques.

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Prove x <=> y is logically equivalent to (x-->y) ^ ((~x)-->(~y)).
 
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aorick21 said:
Prove x <=> y is logically equivalent to (x-->y) ^ ((~x)-->(~y)).

Please show some sort of work or at least tell us where you are stuck. We help with your homework, not do your homework.
 
Actually, x \Leftrightarrow y is shorthand for a longer expression. Which one? Now rewrite one of the subexpressions and you're done.

If you want more specific help, please refer to l46kok's post.

Also, how specific do you need the proof to be? Can you use "intuitive" rules or do you really have to produce a proof tree?
 
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I'd just use a truth table.

If you don't know what that is, then I don't think you belong in math.
 
TimNguyen said:
I'd just use a truth table.

If you don't know what that is, then I don't think you belong in math.

Rubbish. If you do know what one is then perhaps you belong in computer science or electrical engineering?
 
Aoik: what is the contrapositive of ~x=>~y

p.s.: Matt, you crack me up.
 

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