permit \$A = 1,2,3\$ and \$B = 7,8,9\$. For the relation between \$A\$ and \$B\$ given as a subset the \$A imes B = (1,9), (1,7), (3,8)\$, \$A imes B\$ is no a role because \$1\$ appears as the an initial element in an ext than one ordered pair. This make me think around the complying with generalized question.

If a relationship is not a function, climate it is just a subset the \$A\$ and also \$B\$ denoted together \$A imes B\$. Then have the right to this subset be one-to-one? can it it is in onto?

One-to-One?: also though \$A imes B\$ is not a function, I believe it satisfies the requirements of being one-to-one due to the fact that in \$A\$, \$x_1 = x_2\$ only as soon as \$x_1 = x_1\$.

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Onto?: I want to speak yes also. The range of \$A\$ might be the selection of \$B\$. If \$A imes B = (1,9), (1,8), (3,8)\$, then does this satisfy the need of gift onto?

Basically what is piquing mine curiosity is if non-function relations have the right to be one-to-one or onto or if gift one-to-one and/or ~ above implies the it should be a function.

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request Mar 17 "19 at 17:50 Evan KimEvan Kim
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The answer right here is yes, relationships which space not attributes can likewise be defined as injective or surjective. You deserve to read an ext about the on the wikipedia page below under the section titled "Special species of binary relations," yet the interpretations are the organic ones:

A relationship \$Rsubset A imes B\$ is injective if for every \$a,a"in A\$ and \$bin B\$, \$(a,b)in R\$ and also \$(a",b)in R\$ means \$a = a"\$.

A relationship \$Rsubset A imes B\$ is surjective if for each \$bin B\$, over there is part \$ain A\$ such that \$(a,b)in R\$.

Some examples will hope clarify:

If \$A = \ubraintv-jp.combf1\$, a singleton through one element, and also \$B = \ubraintv-jp.combf 1,ubraintv-jp.combf 2,ubraintv-jp.combf 3\$, climate the relationship \$R = (ubraintv-jp.combf 1,ubraintv-jp.combf 1), (ubraintv-jp.combf 1,ubraintv-jp.combf 2), (ubraintv-jp.combf 1,ubraintv-jp.combf 3)\$ is both injective and also surjective, and also it is not a function.

If \$A = \ubraintv-jp.combf 1,ubraintv-jp.combf 2,ubraintv-jp.combf 3\$, \$B = \ubraintv-jp.combf 1,ubraintv-jp.combf 2\$, climate the relationship \$R = (ubraintv-jp.combf 1,ubraintv-jp.combf 1), (ubraintv-jp.combf 2,ubraintv-jp.combf 1)\$ is not injective, no surjective, and also it is no a function.

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If \$A = Bbb R\$, the set of genuine numbers, \$B = Bbb R_ge 0\$, the set of non-negative actual numbers, and \$Rsubset A imes B\$ is the relation \$R=(x,x^2): xin Bbb R\$, then \$R\$ is a surjective function, but it is not injective.