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Let $m, n$ be positive integers. Let $X$ be a non-empty set.

a. If $m$ is the less than or equal to n, find an injective map $f: X^m \rightarrow X^n$

b. Find a bijective map $g: X^m \times X^n \rightarrow X^{m+n}$.

I'm just looking for information on what exactly the question is asking. I figured for A the question is asking for a map from an element $X_i$ in $X^m$ onto the matching $X_i$ in $X^n$, since the question is only asking for injectivity and doesn't require the map to span $X^n$. I'm sorry if my formatting is confusing.

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    I tried reformatting your question in what I thought you meant, does it look correct? Sorry if it isn't.2011-07-20
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    Yea, that looks better. Thanks.2011-07-20
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    Can you think of an injective map $\mathbf{R} \to \mathbf{R}^2$?2011-07-20
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    You're on the right track! You should be explicit about what you mean by "the matching $X_i$ in $X^n$," because there are lots of potential matching $X_i$.2011-07-20
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    I will answer (a). Let $k$ be a certain element of $X$, fixed from now on. Map any $m$-tuple $(x_1,\cdots,x_m)$ in $X^m$ to the $n$-tuple $(x_1,\dots,x_m,k,\dots,k)$ where there are $n-m$ $k$'s. It is easy to show that this mapping is injective. Do this for $X$ your favourite set, $m=3$, $n=5$ to make sure you understand. Your proof, though based on the right intuition, was too vague.2011-07-20

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