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

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The discussion focuses on proving the statement that if A is less than or equal to B and B is less than or equal to A, then A must equal B. It references the trichotomy law, which asserts that for any two numbers A and B, one of three conditions must hold: A is greater than B, A is less than B, or A equals B. The participants clarify that since A is less than or equal to B, A cannot be greater than B, and since B is less than or equal to A, A cannot be less than B. The equivalence of the trichotomy law to the expression A < B or B < A or A = B is also questioned. The discussion emphasizes the logical structure underpinning the proof of equality based on the given inequalities.
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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:
$A<B\vee B<A\vee A=B$
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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