Title
THE MATHEMATICS OF FUZZINESS: REALITIES AND BEYOND: Randomness Fuzziness Consistency Principle,Used
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Imposing a probability law on an interval in which a normal fuzzy number has been defined, and then trying to find consistency between randomness and fuzziness is not logical. If we need to establish a probability law followed by a random variable defined in a given interval, there are mathematical formalisms in the theory of statistical inferences to do so. Instead, defining a normal fuzzy number around a point, and then using a conversion factor to deduce a probability density function from the fuzzy membership function is against statistical norms. Probability densities are not found in this way. In fact, the left reference function of a normal fuzzy number is a probability distribution function, and the right reference function is a complementary probability distribution function. Hence, we need two probability laws to define the membership function of a normal fuzzy number. That the membership function is expressible as a distribution function and a complementary distribution function on the left and on the right respectively of the value with unit membership should be the real randomness fuzziness consistency principle.
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Shipping & Returns
Shipping
We ship your order within 2–3 business days for USA deliveries and 5–8 business days for international shipments. Once your package has been dispatched from our warehouse, you'll receive an email confirmation with a tracking number, allowing you to track the status of your delivery.
Returns
To facilitate a smooth return process, a Return Authorization (RA) Number is required for all returns. Returns without a valid RA number will be declined and may incur additional fees. You can request an RA number within 15 days of the original delivery date. For more details, please refer to our Return & Refund Policy page.
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Warranty
We provide a 2-year limited warranty, from the date of purchase for all our products.
If you believe you have received a defective product, or are experiencing any problems with your product, please contact us.
This warranty strictly does not cover damages that arose from negligence, misuse, wear and tear, or not in accordance with product instructions (dropping the product, etc.).
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Your payment information is processed securely. We do not store credit card details nor have access to your credit card information.
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Visa, MasterCard, American Express, Paypal, Shopify Payments, Shop Pay and more.