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1- Let $X = \{1,2,3,7,12\} $ and $Y = \{1,15,7,4,20\} $. We use notation $(x,y)$ to denote that the element $x \in X$ is assigned to (or paired with) the element $y \in Y$. For the relations defined below answer the following questions:

  • Does the relation define a function from $X$ to $Y$?
  • If it does not define a function, explain why not.
  • If the relation defines a function, decide whether the function is injective or not and explain why. Also decide whether the function surjective or not and explain why.

$$\begin{align} (a)&\Big\{ (1,15),(7,7),(3,7),(12,4) \Big\}\\ (b)&\Big\{ (1,1),(3,4),(7,7) \Big\}\\ (c)&\Big\{ (1,15),(3,7),(7,4),(12,20) \Big\}\\ (d)&\Big\{ (1,4),(3,7),(7,1),(1,15) \Big\} \end{align}$$

2- Let $g \, : X \to Y$ be a function. Suppose $A \subseteq X$, that is $A$ is a subset of or equal to $X$. Suppose $B=g(A) \subseteq Y$. Answer the following questions:

$\begin{align} (a)&\text{If}\, x \in A, \text{what can you say about}\, g(x)?\\ (b)&\text{If}\, y \in g(A), \,\text{what does this mean?}\\ (c)&\text{If}\, x \in X \, \text{and}\, g(x) \in g(a), \,\text{is it necessarily true that}\, x \in A? \end{align}$

I just have a few issues.

For 2. I said all of those except (d)[as it is not a function ] are not surjective since there is always one element in Y that is not paired. Is that right?

For 3.

a) g(x)=B?

b) A=x.

c) Is not always true. It is not the case for functions that are not injective.

I am really having doubts about question 3 ( all of it ). Are my answers correct? Many thx in advance.

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1 Answer 1

up vote 1 down vote accepted

(b) is not a function $X\to Y$ either as $12$ is not mapped anywhere. (c) is injective and surjectivity is indeed already not possible because $Y$ has more elements than $X$.

3.(a). No. $g(x)\in B$. (b) No. $y\in B$ or more interestingly, there exists $x\in A$ with $y=g(x)$. (c) You are right

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