The Math Thought Mindset
Logic, Reasoning, and Proofs
ISBN 9781032959771
352 pp
Table of Contents
Ch. 1Logic Basics
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1.1Introduction
1.2Basic Logical Operators
1.3Logical Equivalences
1.4Conditional Statements
1.5Universal & Existential Quantifiers
1.6Negating Quantifiers
Chapter Challenge
Which of the following correctly expresses the negation of ∀x, P(x)?
Correct. The negation of ∀x, P(x) is ∃x, ¬P(x). The universal claim says ‘everything satisfies P’; to refute it, you only need one counterexample — one of the most commonly misunderstood rules in Chapter 1.
Not quite. Negating a quantifier flips it: ∀ becomes ∃, and the predicate gets negated too. To disprove ‘all swans are white’, you need one swan that is not white.
Ch. 2Deductive Reasoning
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2.1Introduction
2.2Valid Argument
2.3Rules of Inference
2.4Quantified Rules of Inference
2.5Heuristic Diagrams
Chapter Challenge
An argument is valid if and only if:
Exactly. Validity is about the logical relationship between premises and conclusion — not whether either is actually true. It guarantees truth-preservation, not truth itself.
Close, but not quite. Validity is a structural property, not a factual one. An argument can be valid with entirely false premises. The test: can the conclusion ever be false if the premises are true?
Ch. 3Elementary Set Theory
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3.1Introduction
3.2Set Fundamentals
3.3Set Operations
3.4Element Chasing
3.5Set Identities and Laws
3.6The Pigeonhole Principle
Chapter Challenge
To prove A ⊆ B by element chasing, the correct structure is:
Right. Element chasing for A ⊆ B always begins: ‘Let x be an arbitrary element of A.’ Then you show x must also be in B. This is Chapter 3’s core technique.
Not quite. To prove containment A ⊆ B, you start from A, not B. Take an arbitrary x ∈ A and track it forward to show it lands in B. The direction matters.
Ch. 4Proof Methods
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4.1Introduction
4.2Grammar and Fundamentals
4.3Quantified Statements
4.4Direct Proofs
4.5Indirect Proofs: Contradictions
4.6Indirect Proofs: Contrapositive
4.7Mathematical Induction
4.8Strong Mathematical Induction
4.9Other Proof Methods
4.10Developing Conjectures
Chapter Challenge
The contrapositive of ‘If P then Q’ is:
Perfect. The contrapositive flips and negates both parts: ¬Q → ¬P. A conditional and its contrapositive are logically equivalent — proving one proves the other.
Not quite. The contrapositive is not the converse. You flip the direction AND negate both: P → Q becomes ¬Q → ¬P. The converse and inverse are different — and neither is equivalent to the original.
Ch. 5Boolean Algebra
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5.1Introduction
5.2Boolean Algebra
5.3Canonical Forms: SOP and POS
5.4Karnaugh Maps
5.5Digital Circuits
5.6Universal Gates
Chapter Challenge
In Boolean algebra, which gate is universal — meaning any other gate can be built from it alone?
Correct. The NAND gate is universal: NOT, AND, OR, and any other logical function can be built from NAND gates alone. NOR is also universal — Chapter 5.6 explores why.
Not quite. While AND and OR are fundamental, neither alone can produce all Boolean functions (you can’t get NOT from AND or OR alone). NAND and NOR are the universal gates.
Appendix
A.1 — Rules of Inferences
A.2 — Quantified Rules of Inferences
A.3 — Logic and Venn Diagrams
A.4 — Well-Ordering Principle
A.5 — List of Commonly Used Symbols
