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The complement of the intersection of two sets is the same as the union of their complements; or not (A or B) = (not A) and (not B) not (A and B) = (not A) or (not B) where "A or B" is an "inclusive or" meaning at least one of A or B rather than an "exclusive or" that means exactly one of A or B. De Morgan's law with set subtraction operation
So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist. However, when restricted to the context of subsets of a given fixed set X {\displaystyle X} , the notion of the intersection of an empty collection of ...
The intersection of A with any of B, C, D, or E is the empty set. In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their intersection is the point at
In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set that is not empty (where by definition, "is empty" means that the statement () is true) might not have an inhabitant (which is an such that ).
For example, {1, 2} is a subset of {1, 2, 3}, and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B.
In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality (+) = + is always true in elementary algebra. For example, in elementary arithmetic , one has 2 ⋅ ( 1 + 3 ) = ( 2 ⋅ 1 ) + ( 2 ⋅ 3 ) . {\displaystyle 2\cdot (1+3)=(2\cdot 1)+(2\cdot 3).}
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When the intersection of a sphere and a plane is not empty or a single point, it is a circle. This can be seen as follows: Let S be a sphere with center O, P a plane which intersects S. Draw OE perpendicular to P and meeting P at E. Let A and B be any two different points in the intersection.