The difference between Filter and Ideal
When used as nouns, filter means a device which separates a suspended, dissolved, or particulate matter from a fluid, solution, or other substance, whereas ideal means a perfect standard of beauty, intellect etc., or a standard of excellence to aim at.
Filter is also verb with the meaning: to sort, sift, or isolate.
Ideal is also adjective with the meaning: optimal.
check bellow for the other definitions of Filter and Ideal
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Filter as a noun:
A device which separates a suspended, dissolved, or particulate matter from a fluid, solution, or other substance; any device that separates one substance from another.
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Filter as a noun:
Electronics or software that separates unwanted signals (for example noise) from wanted signals or that attenuates selected frequencies.
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Filter as a noun:
Any item, mechanism, device or procedure that acts to separate or isolate.
Examples:
"He runs an email filter to catch the junk mail."
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Filter as a noun (figurative):
self-restraint in speech.
Examples:
"He's got no filter, and he's always offending people as a result."
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Filter as a noun (mathematics, order theory):
A non-empty upper set (of a partially ordered set) which is closed under binary infima (a.k.a. meets).
Examples:
"The collection of cofinite subsets of ''ℝ'' is a filter under inclusion: it includes the intersection of every pair of its members, and includes every superset of every cofinite set."
"If (1) the universal set (here, the set of natural numbers) were called a "large" set, (2) the superset of any "large" set were also a "large" set, and (3) the intersection of a pair of "large" sets were also a "large" set, then the set of all "large" sets would form a filter."
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Filter as a verb (transitive):
To sort, sift, or isolate.
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Filter as a verb (transitive):
To diffuse; to cause to be less concentrated or focused.
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Filter as a verb (intransitive):
To pass through a filter or to act as though passing through a filter.
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Filter as a verb (intransitive):
To move slowly or gradually; to come or go a few at a time.
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Filter as a verb (intransitive):
To ride a motorcycle between lanes on a road
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Ideal as an adjective:
Optimal; being the best possibility.
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Ideal as an adjective:
Perfect, flawless, having no defects.
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Ideal as an adjective:
Pertaining to ideas, or to a given idea.
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Ideal as an adjective:
Existing only in the mind; conceptual, imaginary.
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Ideal as an adjective:
Teaching or relating to the doctrine of idealism.
Examples:
"the ideal theory or philosophy"
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Ideal as an adjective (mathematics):
Not actually present, but considered as present when limits at infinity are included.
Examples:
"'ideal point"
"An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle."
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Ideal as a noun:
A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.
Examples:
"Ideals are like stars; you will not succeed in touching them with your hands. But like the seafaring man on the desert of waters, you choose them as your guides, and following them you will reach your destiny'' - [[w:Carl Schurz Carl Schurz]]"
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Ideal as a noun (algebra, ring theory):
A subring closed under multiplication by its containing ring.
Examples:
"Let <math>\mathbb{Z}</math> be the ring of integers and let <math>2\mathbb{Z}</math> be its ideal of even integers. Then the quotient ring <math>\mathbb{Z} / 2\mathbb{Z}</math> is a Boolean ring."
"The product of two ideals <math>\mathfrak{a}</math> and <math>\mathfrak{b}</math> is an ideal <math>\mathfrak{a b}</math> which is a subset of the intersection of <math>\mathfrak{a}</math> and <math>\mathfrak{b}</math>. This should help to understand why maximal ideals are prime ideals. Likewise, the union of <math>\mathfrak{a}</math> and <math>\mathfrak{b}</math> is a subset of <math>\mathfrak{a + b}</math>."
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Ideal as a noun (algebra, order theory, lattice theory):
A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).
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Ideal as a noun (set theory):
A collection of sets, considered small or negligible, such that every subset of each member and the union of any two members are also members of the collection.
Examples:
"Formally, an ideal <math>I</math> of a given set <math>X</math> is a nonempty subset of the [[powerset]] <math>\mathcal{P}(X)</math> such that: <math>(1)\ \emptyset \in I</math>, <math>(2)\ A \in I \and B \subseteq A\implies B\in I</math> and <math>(3)\ A,B \in I\implies A\cup B \in I</math>."
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Ideal as a noun (algebra, Lie theory):
A Lie subalgebra (subspace that is closed under the Lie bracket) 𝖍 of a given Lie algebra 𝖌 such that the Lie bracket [𝖌,𝖍] is a subset of 𝖍.