Commutativity

Commutativity
Tag	`04E4V9G5FR`
URI	`m2://04E4V9G5FR`
Kind	definition
Writing	W1 stated
MSC	08A02
Defines	Commutativity
Uses	Binary operation

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Definition

Let $S$ be a set and let $\ast$ be a binary operation on $S$ . The operation $\ast$ is commutative when $a \ast b = b \ast a$ for all $a, b \in S$ .

Remarks

Addition and multiplication of real numbers are commutative. Subtraction is not commutative: $1 - 2 \neq 2 - 1$ . Multiplication of $2 \times 2$ real matrices is associative but not commutative.

Explanations

L1 human

Commutativity says that the order of the two inputs does not matter. For example, $3 \times 5$ and $5 \times 3$ are both $15$ . Many operations do not have this property, for example subtraction and the composition of rotations in space.

What this node needs for W2

edge instance_of in: one example (0 of 1)
edge not_instance_of in: one non-example that fails one clause (0 of 1)

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What this needs

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Binary operation definition
Commutativity definition

What depends on this

23 nodes use this node, directly or through other nodes.

Extensions of solvable groups are solvable 01EY0EM7RP
Quotients of solvable groups are solvable 027ZQS04SJ
Infinite cyclic group 02B1T50JD1
Circle group 04EAFVFTF7
Cyclic groups are abelian 04RG02S6BA
Subgroups of solvable groups are solvable 05Y0F9M0TS
Abelian group 071DDFVG4J
Classification of finite abelian groups 07CQR9BW3R
Additive group of integers 0BSCJA5G53
Burnside's p^a q^b theorem 0FMAQSB1SC
Nilpotent groups are solvable 0GNG2KVPZJ
The symmetric group on five or more letters is not solvable 0J53NE674D
Subgroups of abelian groups are normal 0JPDTEDCFD
A finite abelian group is cyclic iff its exponent equals its order 0MC0AQZSHE
Solvable group 0P3KZK0FTV
Klein four-group 0PMRQJVTME
Abelian groups are solvable 0R0X73XHAT
Torsion subgroup 0RH1TPDGCQ
Structure theorem for finitely generated abelian groups 0RY0B72XAS
Subgroups of the integers 0SRATCFF1W
Group of nth roots of unity 0WX11EA3F3
Integers modulo n as a quotient group 0YJ3WRR9RY
Free abelian group 0YMSEZE1VH