Exercise 4.1.1 Verify that the definition of equality on the integers is both reflexive and symmetric.
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\(\bullet\) Show that \(a—b= a—b\)
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\(\Vdash\) \(a + b = a + b\)
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\(\equiv\) { Definition 4.1.1 }
- \(a—b= a—b\)
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\(\square\)
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\(\bullet\) Show that \(a—b = c—d \implies c—d = a—b\)
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\(\Vdash\) \(a—b = c—d\)
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\(\equiv\) { Definition 4.1.1 }
- \(a + d = c + b\)
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\(\equiv\) { Symmetry of the equality on \(\mathbb N\) }
- \(c + b = a + d\)
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\(\equiv\) { Definition 4.1.1 }
- \(c — d= a — b\)
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\(\square\)