Basic Inequality Prove: $A\leq B\wedge B\leq A \Rightarrow A=B$

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SUMMARY

The discussion focuses on proving the mathematical statement that if \( A \leq B \) and \( B \leq A \), then \( A = B \). This conclusion is derived from the trichotomy law, which asserts that for any two numbers \( A \) and \( B \), one and only one of the following must hold: \( A > B \), \( A < B \), or \( A = B \). The proof confirms that since \( A \leq B \) negates \( A > B \) and \( B \leq A \) negates \( A < B \), the only valid conclusion is \( A = B \).

PREREQUISITES
  • Understanding of basic inequalities in mathematics
  • Familiarity with the trichotomy law in real numbers
  • Knowledge of logical equivalences and implications
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties of inequalities in real numbers
  • Explore the implications of the trichotomy law in mathematical proofs
  • Learn about logical equivalences and their applications in proofs
  • Investigate advanced topics in real analysis related to inequalities
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Mathematics students, educators, and anyone interested in understanding the foundations of inequalities and their proofs in real analysis.

solakis1
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Prove:

$A\leq B\wedge B\leq A\Rightarrow A=B$
 
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Trichotomy: Given numbers A and B, one and only one must be true:
1) A> B
2) A< B
3) A= B

Since [math]A\le B[/math], A> B is not true.
Since [math]B\le A[/math], A< B is not true.
 
Is the trichotomy law you are using Equivelant to the following:
$A<B\vee B<A\vee A=B$
 

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