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By Graham J. O. Jameson

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5. Let f : Q −→ Q be a mapping defined by the rule f (x) = 3x − |x|, for x ∈ Q. Is f injective? If yes, find an inverse for f . 6. Let f : Q −→ Q be the mapping defined by f (x) = 2x + |x|, for x ∈ Q. Is f injective? If yes, find an inverse for f . 7. Let f : R −→ R be the mapping defined by f (x) = x2 whenever x ≥ 0, x(x − 3) whenever x < 0. Is f injective? If yes, find an inverse for f . 8. Let f : N × N −→ N be the mapping defined by f (n, m) = 2n−1 (2m − 1). Is f injective? If yes, find an inverse for f .

4 3 2 1 Now consider the set A¯ = {a1 , a2 , a3 , a4 } and define the permutation π¯ on A¯ by the chart a1 a 2 a 3 a 4 ↓ ↓ ↓ ↓ . a4 a3 a2 a1 Evidently, the first chart π could be used to represent the transformation π¯ of the set A¯ so we can represent a transformation of a set by indexing its elements and then tracking the changes in this indexing generated by the transformation. In the same way, a permutation of any indexed set can be represented by the corresponding change in the indices. Since any finite set can be indexed, this gives us an easy way to represent such transformations.

For points (a, b), (c, d) ∈ R2 define (a, b) c2 + d 2 . (c, d) to mean that a2 +b2 = (a) Prove that is an equivalence relation on R2 . (b) List all elements in the set {(x, y) ∈ R2 | (x, y) (0, 0)}. (c) List five distinct elements in the set {(x, y) ∈ R2 | (x, y) (1, 0)}. 20. Two n × n matrices A and B are said to be similar if there exists an invertible n × n matrix P such that P−1 AP = B. Show that similarity is an equivalence relation on Mn (R). 1 SOME PROPERTIES OF INTEGERS: MATHEMATICAL INDUCTION In this chapter, we will consider some basic properties of the set Z of integers.

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