(b) Indicate which of the following statements are true and which are

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Module 1: Sets, Relation and Function.

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(b) Indicate which of the following statements are true and which are false. Justify your answers as best you can:

(i) $\forall x \in \mathbf{Z}^+, \exists y \in \mathbf{Z}^+$ such that $x = y+1$
(ii) $\forall x \in \mathbf{Z}, \exists y \in \mathbf{Z}$ such that $x = y+1$
(iii) $\exists x \in \mathbf{R}$ such that $\forall y \in \mathbf{R}, x = y+1$
(iv) $\forall x \in \mathbf{R}^+, \exists y \in \mathbf{R}^+$ such that $xy = 1$
(v) $\forall x \in \mathbf{R}, \exists y \in \mathbf{R}$ such that $xy = 1$
(vi) $\forall x \in \mathbf{Z}^+ \text{ and } \forall y \in \mathbf{Z}^+, \exists z \in \mathbf{Z}^+ \text{ such that } z = x-y$
(vii) $\forall x \in \mathbf{Z} \text{ and } \forall y \in \mathbf{Z}, \exists z \in \mathbf{Z} \text{ such that } z = x-y$
(viii) $\exists u \in \mathbf{R}^+ \text{ such that } \forall v \in \mathbf{R}^+, uv < v$

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