Week 2 Exercises, due 5 October 2021

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Exercises 2.7 Week 2 Exercises, due 5 October 2021

For the given sets \(X\) and \(Y\) and for the given subset \(\Gamma \subseteq X \times Y\text{,}\) please say whether \(\Gamma\) is the graph of a map \(f \colon X \to Y\text{,}\) and if it is, whether \(f\) is an injection, surjection, bijection (or none of these). Each of these is worth 1 point.

1.

\(X = \NN_0\text{,}\) \(Y = \NN_0\text{,}\) and \(\Gamma = \{(x,y)\in\NN_0 \times \NN_0 : x = y\}\text{.}\)

2.

\(X = \NN_0\text{,}\) \(Y = \NN_0\text{,}\) and \(\Gamma = \{(x,y)\in\NN_0 \times \NN_0 : 2x = y\}\text{.}\)

3.

\(X = \NN_0\text{,}\) \(Y = \NN_0\text{,}\) and \(\Gamma = \{(x,y)\in\NN_0 \times \NN_0 : x = 2y\}\text{.}\)

4.

\(X = \NN_0\text{,}\) \(Y = \NN_0\text{,}\) and \(\Gamma = \{(x,y)\in\NN_0 \times \NN_0 : 2y \leq x \leq 2y+1\}\text{.}\)

5.

\(X = 3\text{,}\) \(Y = 4\text{,}\) and \(\Gamma = 3 \times \{0\}\text{.}\)

6.

\(X = 3\text{,}\) \(Y = 4\text{,}\) and \(\Gamma = \{0\}\times 4\text{.}\)

7.

\(X = 0\text{,}\) \(Y = 0\text{,}\) and \(\Gamma = 0\text{.}\)

8.

\(X = 0\text{,}\) \(Y = 1\text{,}\) and \(\Gamma = 0\text{.}\)

9.

\(X = 1\text{,}\) \(Y = 0\text{,}\) and \(\Gamma = 0\text{.}\)

10.

\(X = 1\text{,}\) \(Y = 1\text{,}\) and \(\Gamma = \{(0,0)\}\text{.}\)