Subset : Superset
If X and Y are sets and every element of X is also an element of Y, then we say or write:
- X is a subset of Y;
- X ⊆ Y;
- Y is a superset of X;
- Y ⊇ X.
Every set Y is a subset of itself. A subset of Y which isn't equal to Y is called proper. If X is a proper subset of Y, then we write X ⊂ Y. Analogous comments apply to supersets.
Notational variations
There are two major systems in use for the notation of subsets. The older system uses the symbol "⊂" to indicate any subset and uses "⊊" to indicate proper subsets. The newer system uses the symbol "⊆" to indicate any subsets and uses "⊂" to indicate proper subsets. Wikipedia uses the newer system, which can be handled by a wider variety of web browsers. Analogous comments apply to supersets.
Examples
- The set {1,2} is a proper subset of {1,2,3}.
- The set of natural numbers is a proper subset of the set of rational numbers.
- The set {x : x is a prime number greater than 2000} is a proper subset of {x : x is an odd number greater than 1000}
- Any set is a subset of itself, but not a proper subset.
- The empty set, written {}, is also a subset of any given set Y. (This statement is vacuously true.) The empty set is always a proper subset, except of itself.
there was no word from beyond the walls. "We must knock harder," said
live," to the words "dangerously ill." At length, on Wednesday, 6th May,
Desault to give the invalid the assistance of his art. After having
He made a long and very attentive examination of the unfortunate child.html">child,
contented himself with prescribing a decoction of hops, to be taken by
the evening. On the first day the Prince steadily refused to take it.
presence; his example proved as ineffectual as his words. Next day Lasne
nothing but the good of his health.html">health, and he distresses me deeply by thus
not to give me this cause of grief." And as Lasne, while speaking, began
his hands. "You have, then, taken an oath that I should drink it," said
conformed with docility to whatever was required of him, but the policy
was almost a mockery to supply it.
The Prince's weakness was excessive; his keepers could scarcely drag him
he stopped to press the arm of Lasne with both hands upon his breast. At
and his keeper carried him about, sometimes on the platform.html">platform, and
But the slight improvement to his health occasioned by the change of air
battlement of the platform nearest the left turret, the rain had, by
fell remained there for several days; and as, during the spring of 1795,
constantly supplied. Whenever the child was brought out upon the
and bathe in this reservoir. At first they flew away at his approach,
they at last grew more familiar, and did not spread their wings for
them by sight, and perhaps like himself they were inhabitants of that
.
On
wordlookup.net
All is still licensed under the GNU FDL.
It uses material from the wikipedia.
|
|