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For any two points P=(p(1),p(2),…,p(n)) and Q=(q(1),q(2),…,q(n)) of ℝn, we define the crispvector PQ⟶=(q(1)−p(1),q(2)−p(2),…,q(n)−p(n))=Q(−)P. Then we obtain an n-dimensional vector space En={PQ⟶|   for all   P,Q∈ℝn}. Further, we extend the crisp vector into the fuzzy vector onfuzzy sets of ℝn. Let D˜,E˜ be any two fuzzy sets on ℝn and define the fuzzy vector E˜D˜⟶=D˜⊖E˜, then we have a pseudo-fuzzy vector space

international journal of mathematics and mathematical sciences

Extension of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$n$"><mml:mi>n</mml:mi></mml:math>-dimensional Euclidean vector space<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$E^n$"><mml:msup><mml:mi>E</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math>over<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$\mathbb{R}$"><mml:mi>ℝ</mml:mi></mml:math>to pseudo-fuzzy vector space over<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext…

Extension ofn-dimensional Euclidean vector spaceEnoverℝto pseudo-fuzzy vector space over