Sets in `Logic`

 

The `Logic` Set concepts of�

            �unchanging�, �not�, �intersection�, �union�, and �complement��

 ï¿½are implemented with the LogicalPatternSets�

 

            True[set], Not[set], And[�sets�], Or[�sets�], and Xor[�sets�] respectively.

If set is any PatternSet , then:

True[set] matches elements in set having unassigned values.

False[set] matches elements in set having assigned values.

Not[set] matches everything that is not in set.

If ��sets�� is any sequence of PatternSets, then:

And[�sets�] is the intersection of ��sets...�.

Or[�sets�] is the union of all of the ��sets��.

Xor[set1, �sets�] matches elements from set1 found in zero or any even number of elements in ��sets��.

 

 

 

In the following, if sets is a PatternSet, then

 

(1)                                True[set]

 

�is the subPatternSet of all unassigned elements in set.

 

Similarly,

 

                                    Not[True][set]

(2)                    or

                                    False[set]

 

�is the subPatternSet of all elements in set, that have assigned values.  In other words, the Set of expressions that change when Reckoned.

 

False[set] is the PatternSet whose elements have assigned values.  If "False[set][el]" is True, el has an assigned value.

 

 

 

 

If set is a PatternSet,

 

(3)                    Not[set]

 

�is the PatternSet containing everything that is not in �set�.

 

If el is any Expression, Not[set[el]] is equivalent to Not[set][el].

 

 

 

 

If ��sets�� is one or more PatternSet(s), then�

 

(4)                    And[�sets�]

 

�matches any element in the intersection of these �sets�.

(4) matches any element in all of the �sets�.

 

 

 

If ��sets�� is one or more PatternSet(s), then�

 

(5)                    Or[�sets�]

 

�matches any element in the union of the �sets�.

 

 

 

If set is a PatternSet, then Xor[set] is equivalent to set.

 

When Xor[�] is parameterized by more the one PatternSet, it implements set complement.

If set1 is any PatternSet, and �sets� is one or more PatternSet, then�

 

(6)                    Xor[set1, �sets�]

 

�matches elements in set1 that are found in zero or any even number of elements in �sets�.

 

 

 

 [A COMPLEMENT implies two things which mutually complete each other and together constitute a whole.]

Complement n.

1. a That which fills up or completes.

    b The quantity or number required to fill a thing or make it complete; full allowance. 

    c That which is required to supply a deficiency, to make perfect, or to complete a symmetrical whole; one of two mutually completing parts or fact of being complete.

 

[The above definition is consistent with the Complement-Xor function in Grok32`.  See 8.d. below.]

 

2. Act of completing or fulfilling; accomplishment; state or fact of being complete.  Obs.

3. Obs.

   a. Something added for ornamentation; an accessory. 

   b A quality or accomplishment or a �finished� gentleman. 

   c A ceremonial observance; a compliment.

4. Full quantity, number, or amount; a complete set.

            To exceed his complement and number appointed him, which was one hundred and twenty persons.

5. Grammar.

   a. Any added word or words by which a predication is made complete (they made Saul king; to set the slaves free; filled them with terror).

   b. Such a word or words filling out the sense of a verb of incomplete predication (it proved of no use).

6. Her. Fullness (of the moon).

�

8. Math. 

   a The amount of angle or arc by which a given angle or arc falls short of 90�.

   b Of a logarithm, the numerical amount by which the logarithm falls short of 10.

   c Of a number, the numerical amount that must be added to the number to give the least number containing one more digit; as, the complement of 4 is 6, and that of 45 is 55.

[  d. Boolean Logic. The set with elements from set1 found in zero or any even number of other sets.  This set is called the complement to set1.  (In this context,) the �Exclusive-Or� function, (Xor[set1 , sets]), and the complement are the same.  See 1.c. above.]

9. Music.  The interval required with a given interval to complete the octave.  See INTERVAL.

10. Naut.  The whole force or personnel of a vessel; specif., Nav., the entire force of officers and crew allowed to a ship.

 

A COMPLEMENT is that which makes up or supplies a deficiency; it frequently implies two things which mutually complete each other and together constitute a whole; as, Norman refinement and flexibility are the complement, in the English vocabulary, of its Saxon homeliness and strength.  SUPPLEMENT, as here compared (cf. APPENDIX), implies an addition to that which has already relative completeness; it does not imply a mutual dependence on its parts; as, foreign travel is a profitable supplement to a university education.

 

intersection n.

1. Act, state, or place of intersection.

2. Math.

   a The extent (of points, lines, surfaces, in which two other extents (like or unlike) intersect, or which is common to both.  The intersection of two curves consist of points; of two surfaces, of curves; of a line and a surface, in general of points; of a surface and a volume, in general of a surface; etc. 

   b The aggregate of elements in common to two aggregates.

 

intersect v.t.

To pierce or divide by passing through or athwart; to cut across; to cross; as, any two diameters of a circle intersect each other.  �v.i.  To cut into one another; to meet and cross at a point, as two lines.

 

 

union n.

1. Act or instance of uniting or joining two or more things into one; state of being so united; junction; coalition; combination.

2. a A spiritual uniting to bring about concord or agreement; also, the unity so produced.

            The union, peace, and plenty of the kingdom.                   Clarendon

   b In mystical language, the conscious identification of one�s will with that of God; the third and highest stage in mystical striving in which the soul has experimental knowledge of God; --called also the unitive way.

3. A uniting into a nation, political body, or association; specif., any of various political changes affection the political union of the different parts of the kingdom of Great Britain and Ireland; as: that of 1535, uniting Wales to England; that of 1706 , uniting the kingdoms of England and Scotland on and after May 1, 1707; or that of 1800, on and after Mmay 1, 1707; or that of 1800, establishing a legislative union of Greate Britain and Ireland on and after January 1, 1801; also, that of 1909 uniting by the South Africa Act the colonies ofr Sout Africa.

4. A uniting in marriage.

5. a That which is united, or made one; something formed by combination or coalition of parts or members; a confederation ; consolidated body; a league; as, the United States of America are often called �the Union.�  Cf. POSTAL UNION, ZOLLVEREIN.  b Short for UNION OF SOVIET SOCIALIST REPUBLICS.

6. A device�

7. Eng.

   a union of local administration for rating, poor relief, etc., formed by uniting two or more parishes under the government of a body called the board of guardians.  Their formation oand government are now chiefly regulated by the Local Government Act of 1894.

   b A workhouse maintained by such a union.

8. A league or association, esp. a trade-union.

9.  An organized social center of a college or university; also, the building in which it usually is housed;  -- from the name of a debating club at Oxford.

10. Obs. a Oneness.  b A unicum.

Syn.  � UNION, UNITY, UNION suggests either the act of bringing two or more things together, or the resulting state; UNITY denotes state of being one, or property of oneness; and intimate union in a complex whole; as, a union of interests resulting in a unity of aim:  �Such harmony alone could hold all heaven and earth in happier union.� (Milton);  �How good � it is for brethren to dwell together in unity� (Tennyson);  �It was but unity of place that made me dream I ranked with him� (id.).

Cf. COMBINATION, JOIN.

                                                            [From Websters1949Unabridged.]

 

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by John Van Wie Bergamini

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