Concept

Intersection (set theory)

Overview

In set theory, the intersection of two sets

{\displaystyle A}

and

{\displaystyle B,}

denoted by

∩

{\displaystyle A\cap B,}

is the set containing all elements of

{\displaystyle A}

that also belong to

{\displaystyle B}

or equivalently, all elements of

{\displaystyle B}

that also belong to

{\displaystyle A.}

The notion of intersection as an algebraic operation with sets as operands has been generalized from geometry, where it is encountered in the case of geometric sets of points, such as individual points, lines (infinite uncountable sets of points), planes, etc.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

In the original texts

Passages from public-domain books where Intersection (set theory) is discussed.

Furthermore if C be a fourth moment, then apart from special cases which we need not consider, it intersects M in a plane which the straight line (A, B, M) meets. Thus there is in general a common intersection of four moments of different families. This common intersection is an assemblage of abstractive elements which are each covered (or 'lie in') all four moments. The three-dimensional property of instantaneous…
Alfred North Whitehead, The Concept of Nature The Tarner Lectures Delivered in Trinity College, November 1919
names the idea; states what it is; opens on it
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We might also employ the distances from the movable point to three fixed points, which would determine it by the intersection of three spheres with a common centre. In like manner, the position of a point would be defined by giving its distance from a fixed point, and the direction of that distance, by means of the two angles which this right line makes with two invariable axes;…
Auguste Comte, The philosophy of mathematics
names the idea; opens on it
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Two sets of four points each are defined as having the same anharmonic ratio, when (1) each set of four lies in one straight line, and (2) corresponding points of different sets lie two by two on four straight lines through a single point, or when both sets have this relation to any third set[123]. And reciprocally: Two sets of four straight lines are defined as having the same anharmonic ratio when (1) each set of…
Bertrand Russell, An essay on the foundations of geometry
states what it is
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That the squares of the periodic times of the planets are proportional to the cubes of their distances from the sun, is a uniformity derived from the laws of the causes (or forces) which produce the planetary motions; but that the square of any number is four times the square of half the number, is true independently of any cause. The only laws of resemblance, therefore, which we are called upon to consider…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
states what it is
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Thus, instead of an existential world that is "a network of relations whose intersections are called terms" (p. 622), I find more intelligible a qualitatively heterogeneous reality that can be variously partitioned into things, and that can he abstractly replaced by systems of terms and relations that are adequate to symbolize their effective nature in particular respects. There is a tendency for certain attributes…
John Dewey, Creative Intelligence
states what it is; opens on it; names the idea
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Philosophers who discussed Intersection (set theory)

Alfred Tarski

Related ideas

Place in the tradition

Part of Description logic.

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