“Usually, a set is determined by a mathematical statement that describes a property that all members of the set share.” Use set-builder notation to define the set of rational numbers and the set of irrational numbers.
Section 9.2
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Proof technique, just below Definition 9.2.1. In addition, how would you prove that two sets are the same, using the concept of subsets?
Section 9.3
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"... using the logical equivalence of P ⇒ (Q1 ∨ Q2) and (P∧ ∼Q1) ⇒ Q2." Prove that this statement is true.
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Theorem 9.3.7. Prove that these statements are true.
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"Assume y ∈ ℝ, then y ∈ {y, 7, 9}." Explain why this is true.
Section 9.4
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Equation (9.35). Explain why (a, a) = ∅.
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Equation (9.36). Explain why [a, a] = {a}.
Example 9.4.3. Remind yourself of how to prove that two sets are equal.
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First comment below Equation (9.38). Write down the Archimedean Principle in mathematical terms. Check that it is correctly applied in the argument that follows.
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Equation (9.41). Re-write the right-hand side of this inequality in a way that immediately proves it is strictly less than 2.
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First line. Why is it important to state that (2 x + 1)/(2 − x) ∈ ℝ?