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It is a definition that: $A$ has cardinality less than or equal to the cardinality of $B$, if there exists an injective function from $A$ into $B$.
https://mathresearch.utsa.edu/wiki/index.php?title=Sets:Cardinality
Intuitively, it seems true. Can it be proven with some axioms? Or it is just a definition without need to prove it or can't be proven with axioms?

Formal proof for "$A$ has cardinality less than or equal to the cardinality of $B$, if there exists an injective function from $A$ into $B$." such as below proof for the distrubutive law?

enter image description here

Lucenaposition
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    One does not prove definitions. – Clement Yung Sep 02 '24 at 11:49
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    You begin stating: "It is a definition ...". And then you ask "It is just a definition?". Yes, a definition is a definition. – jjagmath Sep 02 '24 at 11:52
  • Definitions in math are very much like definitions in a dictionary, they assign meaning to terminology. In this case, one is assigning meaning to a "less than or equal" relation on cardinalities. On the other hand, verifying particular cases of a definition does require proof. For example: Prove that the cardinality of $\mathbb Z$ is less than or equal to the cardinality of $\mathbb R$. Prove that the cardinality of $\mathbb R$ is not less than or equal to the cardinality of $\mathbb Z$. – Lee Mosher Sep 02 '24 at 13:12
  • hi, do not image rather type in mathjax – TShiong Sep 02 '24 at 13:56
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    This question is similar to: Why do we not have to prove definitions?. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. – Lee Mosher Sep 02 '24 at 13:58

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As said in the comments, there already are great answers to the question “Why don't we have to prove definitions?” in this other question.

Nevertheless, I'd like to clarify one point which is specific to the situation at hand. Normally, you cannot “define $|A| ≤ |B|$” at once, you must define what $|A|$ is and then what $≤$ is. In the case of set theory, defining $|A|$, called the cardinality of $A$, is actually quite difficult to do at an elementary level; see Wikipedia if you're interested in the details, but you most likely won't be able to understand them if you're just getting started with logic and basic undergraduate mathematics.

For this reason, most of the time, when you first meet the notion of cardinality, it is admitted that one can formally define an object $|A|$ (called a cardinal) and a relation $≤$ between cardinals, and it becomes an admitted theorem that $|A| ≤ |B|$ iff there is an injection $A → B$.

Jean Abou Samra
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  • In $ZFC$, you can define "$|A|$ is the least ordinal in bijection with $A$". – Lucenaposition Sep 03 '24 at 02:05
  • @Lucenaposition Yes of course. My point is that the OP seems to be at an early undergrad level so they almost certainly haven't heard about ordinals. Plus you first have to prove the well-ordering theorem. – Jean Abou Samra Sep 03 '24 at 10:15