# Showing the cardinality of A is less than B's

• k3k3
In summary, in order to prove that |A|≤|B| if A and B are sets such that A\subseteqB, we can define a function f:A→B such that f(a)=a for all a in A, which is a 1-1 function. This shows that |A|≤|B| by the given definition of |A|≤|B|. However, if A is a countable subset of B, then |A| = |B|.
k3k3

## Homework Statement

If A and B are sets we say that |A|≤|B| if and only if there exists a one-to-one function f:A→B.

Prove that if A and B are sets such that A$\subseteq$B , then |A|≤|B|.

## Homework Equations

Our text does not define this, so the definition comes from my class notes.

Definition: Suppose A and B are sets.
Then |A|≤|B| iff there exists a 1-1 function from A to B.
Note: Such an an f, if it exists, may or may not be onto. If f is also onto, then |A|=|B|

## The Attempt at a Solution

Since A$\subseteq$B, then either A is B or all of A is in B.

Hence, there exists an injection f such that f:A→B defined by f(a)=a for all a in A.
Since f is not a onto B, then |A| < |B|

If f is a bijection, then, by definition, it follows that |A|=|B|.

Does this proof make sense?

k3k3 said:

## Homework Statement

If A and B are sets we say that |A|≤|B| if and only if there exists a one-to-one function f:A→B.

Prove that if A and B are sets such that A$\subseteq$B , then |A|≤|B|.

## Homework Equations

Our text does not define this, so the definition comes from my class notes.

Definition: Suppose A and B are sets.
Then |A|≤|B| iff there exists a 1-1 function from A to B.
Note: Such an an f, if it exists, may or may not be onto. If f is also onto, then |A|=|B|

## The Attempt at a Solution

Since A$\subseteq$B, then either A is B or all of A is in B.

Hence, there exists an injection f such that f:A→B defined by f(a)=a for all a in A.
Since f is not a onto B, then |A| < |B|

If f is a bijection, then, by definition, it follows that |A|=|B|.

Does this proof make sense?

Just because f is not onto does not imply that |A|<|B|. That's only true for finite sets. Can you give me a counterexample?

|N| < |R| and there is not an onto mapping of N to R

k3k3 said:
|N| < |R| and there is not an onto mapping of N to R

That's not a counterexample. I want an example where f is not onto but |A|=|B|.

I am unsure of an example of that. For infinite sets, all I know is either it will be the same as |N| else it will be what |R| is.

k3k3 said:
I am unsure of an example of that. For infinite sets, all I know is either it will be the same as |N| else it will be what |R| is.

How about A=even integers, B=all integers? Have you looked at cases like that?

If f is a function that takes all integers to even integers, f can be defined as f(n)=2n for all n in the integers. That is onto.

k3k3 said:
If f is a function that takes all integers to even integers, f can be defined as f(n)=2n for all n in the integers. That is onto.

That function would show |A|=|B|. But the function you are using in your proof is f(n)=n. That's is not onto. What I'm trying to point out the that f(n)=n not being onto does not show |A|<|B| which you are saying in your proof.

I see. Then if for all a in A there exists a b in B such that f(a)=b and if A is countable and B is not, f is not onto and |A| < |B|. If A is a countable subset of B, then |A| = |B|.

Is this correct thinking?

k3k3 said:
I see. Then if for all a in A there exists a b in B such that f(a)=b and if A is countable and B is not, f is not onto and |A| < |B|. If A is a countable subset of B, then |A| = |B|.

Is this correct thinking?

Pretty much. But that's not something you need to include in your proof. You defined a 1-1 function from A to B by defining f(a)=a. By your definition that shows |A|≤|B|. That's about all you need to say.

I see. Thank you for the help!

## What does it mean to show the cardinality of A is less than B's?

Showing the cardinality of A is less than B's means that there are fewer elements in set A than in set B. It is a way to compare the sizes of two sets.

## How do you show the cardinality of A is less than B's?

To show the cardinality of A is less than B's, you can use a mathematical proof or a visual representation such as a Venn diagram. The proof should demonstrate that there are more elements in set B than in set A.

## What does it mean for two sets to have the same cardinality?

Two sets have the same cardinality if they have the same number of elements. This means that there is a one-to-one correspondence between the elements of the two sets, and they can be paired up without any elements being left over.

## Can the cardinality of a set be infinite?

Yes, the cardinality of a set can be infinite. This means that the set has an uncountable number of elements. For example, the set of all real numbers is infinite.

## Why is it important to show the cardinality of A is less than B's?

Showing the cardinality of A is less than B's is important because it helps us understand the relationship between two sets. It can also be used to prove other mathematical theorems and to solve problems in various fields such as computer science, statistics, and economics.

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