Find the Number of Subsets of the Set ââ‹math Englishã¢â‹ History Scienceã¢â‹ Art

Set whose elements all vest to another set

Euler diagram showing
A is a subset of B,A⊆B,  and conversely B is a superset of A.

In mathematics, set A is a subset of a fix B if all elements of A are too elements of B; B is and so a superset of A. It is possible for A and B to be equal; if they are unequal, so A is a proper subset of B. The human relationship of one ready beingness a subset of some other is called inclusion (or sometimes containment). A is a subset of B may also be expressed every bit B includes (or contains) A or A is included (or contained) in B.

The subset relation defines a fractional order on sets. In fact, the subsets of a given gear up form a Boolean algebra under the subset relation, in which the bring together and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.

Definitions [edit]

If A and B are sets and every element of A is as well an element of B, then:

If A is a subset of B, simply A is not equal to B (i.e. there exists at to the lowest degree i element of B which is not an chemical element of A), then:

The empty set, written ${\displaystyle \{\}}$ or ${\displaystyle \varnothing ,}$ is a subset of any set Ten and a proper subset of whatever set except itself, the inclusion relation ${\displaystyle \subseteq }$ is a partial club on the set up ${\displaystyle {\mathcal {P}}(S)}$ (the power set of S—the ready of all subsets of S ^[1]) defined by ${\displaystyle A\leq B\iff A\subseteq B}$ . We may also partially order ${\displaystyle {\mathcal {P}}(South)}$ by opposite fix inclusion by defining ${\displaystyle A\leq B{\text{ if and merely if }}B\subseteq A.}$

When quantified, ${\displaystyle A\subseteq B}$ is represented as ${\displaystyle \forall x\left(x\in A\implies 10\in B\correct).}$ ^[2]

Nosotros can prove the statement ${\displaystyle A\subseteq B}$ by applying a proof technique known as the element argument^[three]:

Let sets A and B be given. To prove that ${\displaystyle A\subseteq B,}$

1. suppose that a is a particular simply arbitrarily chosen chemical element of A
2. testify that a is an chemical element of B.

The validity of this technique tin be seen as a result of Universal generalization: the technique shows ${\displaystyle c\in A\implies c\in B}$ for an arbitrarily chosen chemical element c. Universal generalisation and then implies ${\displaystyle \forall 10\left(x\in A\implies 10\in B\correct),}$ which is equivalent to ${\displaystyle A\subseteq B,}$ as stated above.

Properties [edit]

• A set up A is a subset of B if and only if their intersection is equal to A.

Formally:

${\displaystyle A\subseteq B{\text{ if and simply if }}A\cap B=A.}$

• A set A is a subset of B if and only if their wedlock is equal to B.

Formally:

${\displaystyle A\subseteq B{\text{ if and simply if }}A\cup B=B.}$

• A finite set A is a subset of B, if and merely if the cardinality of their intersection is equal to the cardinality of A.

Formally:

${\displaystyle A\subseteq B{\text{ if and simply if }}|A\cap B|=|A|.}$

⊂ and ⊃ symbols [edit]

Some authors apply the symbols ${\displaystyle \subset }$ and ${\displaystyle \supset }$ to point subset and superset respectively; that is, with the same meaning and instead of the symbols, ${\displaystyle \subseteq }$ and ${\displaystyle \supseteq .}$ ^[4] For case, for these authors, it is truthful of every fix A that ${\displaystyle A\subset A.}$

Other authors prefer to use the symbols ${\displaystyle \subset }$ and ${\displaystyle \supset }$ to indicate proper (also called strict) subset and proper superset respectively; that is, with the same meaning and instead of the symbols, ${\displaystyle \subsetneq }$ and ${\displaystyle \supsetneq .}$ ^[v] This usage makes ${\displaystyle \subseteq }$ and ${\displaystyle \subset }$ analogous to the inequality symbols ${\displaystyle \leq }$ and ${\displaystyle <.}$ For example, if ${\displaystyle ten\leq y,}$ then x may or may not equal y, but if ${\displaystyle x<y,}$ then 10 definitely does non equal y, and is less than y. Similarly, using the convention that ${\displaystyle \subset }$ is proper subset, if ${\displaystyle A\subseteq B,}$ then A may or may not equal B, but if ${\displaystyle A\subset B,}$ and so A definitely does not equal B.

Examples of subsets [edit]

The regular polygons grade a subset of the polygons

• The set A = {ane, two} is a proper subset of B = {one, 2, three}, thus both expressions ${\displaystyle A\subseteq B}$ and ${\displaystyle A\subsetneq B}$ are truthful.
• The set D = {1, 2, 3} is a subset (but not a proper subset) of Due east = {1, 2, 3}, thus ${\displaystyle D\subseteq E}$ is true, and ${\displaystyle D\subsetneq Due east}$ is not true (fake).
• Any set is a subset of itself, just not a proper subset. ( ${\displaystyle 10\subseteq X}$ is truthful, and ${\displaystyle Ten\subsetneq X}$ is simulated for whatsoever set X.)
• The ready {x: x is a prime number greater than ten} is a proper subset of {ten: x is an odd number greater than 10}
• The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a line segment is a proper subset of the ready of points in a line. These are 2 examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite ready) as the whole; such cases tin can run counter to one's initial intuition.
• The set of rational numbers is a proper subset of the set of existent numbers. In this example, both sets are infinite, but the latter set has a larger cardinality (or power) than the one-time set.

Some other instance in an Euler diagram:

• 

  A is a proper subset of B

• 

  C is a subset only not a proper subset of B

Other properties of inclusion [edit]

${\displaystyle A\subseteq B}$ and ${\displaystyle B\subseteq C}$ implies ${\displaystyle A\subseteq C.}$

Inclusion is the canonical partial lodge, in the sense that every partially ordered set ${\displaystyle (Ten,\preceq )}$ is isomorphic to some collection of sets ordered past inclusion. The ordinal numbers are a simple example: if each ordinal north is identified with the gear up ${\displaystyle [n]}$ of all ordinals less than or equal to n, then ${\displaystyle a\leq b}$ if and only if ${\displaystyle [a]\subseteq [b].}$

For the power gear up ${\displaystyle \wp {P}(S)}$ of a set up Due south, the inclusion partial order is—upward to an order isomorphism—the Cartesian product of ${\displaystyle k=|South|}$ (the cardinality of S) copies of the fractional club on ${\displaystyle \{0,i\}}$ for which ${\displaystyle 0<1.}$ This tin be illustrated by enumerating ${\displaystyle Due south=\left\{s_{one},s_{two},\ldots ,s_{one thousand}\right\},}$ , and associating with each subset ${\displaystyle T\subseteq S}$ (i.e., each element of ${\displaystyle ii^{South}}$ ) the k-tuple from ${\displaystyle \{0,1\}^{1000},}$ of which the ith coordinate is 1 if and just if ${\displaystyle s_{i}}$ is a member of T.

See too [edit]

• Convex subset
• Inclusion order
• Region
• Subset sum trouble
• Subsumptive containment
• Total subset

References [edit]

1. ^ Weisstein, Eric W. "Subset". mathworld.wolfram.com . Retrieved 2020-08-23 .
2. ^ Rosen, Kenneth H. (2012). Detached Mathematics and Its Applications (7th ed.). New York: McGraw-Loma. p. 119. ISBN978-0-07-338309-v.
3. ^ Epp, Susanna South. (2011). Discrete Mathematics with Applications (Fourth ed.). p. 337. ISBN978-0-495-39132-6.
4. ^ Rudin, Walter (1987), Existent and complex analysis (3rd ed.), New York: McGraw-Colina, p. half dozen, ISBN978-0-07-054234-1, MR 0924157
5. ^ Subsets and Proper Subsets (PDF), archived from the original (PDF) on 2013-01-23, retrieved 2012-09-07

Bibliography [edit]

• Jech, Thomas (2002). Set up Theory. Springer-Verlag. ISBNthree-540-44085-ii.

External links [edit]

• Media related to Subsets at Wikimedia Commons
• Weisstein, Eric W. "Subset". MathWorld.

hubbardwhinged1940.blogspot.com

Source: https://en.wikipedia.org/wiki/Subset

Share this post