Exercise 4.1.3 Show that \((−1) × a = −a\) for every integer \(a\).
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\(\bullet\) Show that \((−1) × a = −a\) for every integer \(a\).
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\(\Vdash\) \((−1) × a\)
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\(=\) { By Definition 4.1.1, \(a = b — c\) for some natural number \(b\) and \(c\); Definition 4.1.4 }
- \((0 — 1) × (b — c)\)
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\(=\) { Definition 4.1.2 }
- \((0b + 1c)—(0c + 1b)\)
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\(=\) { Multiplication on \(\mathbb N\) }
- \((c — b)\)
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\(=\) { Definition 4.1.4 }
- \(−(b — c)\)
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\(=\) { \(a = b — c\) }
- \(−a\)
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\(\square\)