1.2 Logic and Mathematical Reasoning

Logic and mathematical reasoning form the backbone of abstract math. They provide tools to analyze statements, construct valid arguments, and prove theorems. These skills are crucial for understanding complex mathematical concepts and developing rigorous proofs.

This section introduces logical connectives, truth tables, and various proof techniques. It also covers how to analyze mathematical arguments, identify logical fallacies, and construct well-structured proofs. These fundamentals are essential for tackling more advanced topics in abstract mathematics.

Logical Reasoning in Mathematics

Principles and Role in Mathematics

Top images from around the web for Principles and Role in Mathematics

• 

  Logical reasoning - Wikipedia View original

  Is this image relevant?

• 

  Logic and Structure - Basic Reading and Writing View original

  Is this image relevant?

• 

  Logical reasoning - Wikipedia View original

  Is this image relevant?

• 

  Logic and Structure - Basic Reading and Writing View original

  Is this image relevant?

1 of 2

Top images from around the web for Principles and Role in Mathematics

• 

  Logical reasoning - Wikipedia View original

  Is this image relevant?

• 

  Logic and Structure - Basic Reading and Writing View original

  Is this image relevant?

• 

  Logical reasoning - Wikipedia View original

  Is this image relevant?

• 

  Logic and Structure - Basic Reading and Writing View original

  Is this image relevant?

1 of 2

• Logical reasoning derives conclusions from premises or known facts using valid inference rules
• Mathematics relies on logical reasoning to establish the truth of statements and validity of proofs
• Basic principles of logical reasoning include the law of non-contradiction, law of excluded middle, and law of identity
  • Law of non-contradiction states that a proposition cannot be both true and false at the same time ($P \land \neg P$ is always false)
  • Law of excluded middle asserts that a proposition is either true or false, with no other possibilities ($P \lor \neg P$ is always true)
  • Law of identity affirms that a thing is identical to itself ($P \Rightarrow P$ is always true)

Logical Connectives and Truth Tables

• Logical connectives, such as "and" ($\land$), "or" ($\lor$), "not" ($\neg$), "if-then" ($\Rightarrow$), and "if and only if" ($\Leftrightarrow$), form compound statements and express logical relationships between propositions
• Truth tables analyze the truth values of compound statements based on the truth values of their component propositions
  • For example, the truth table for $P \land Q$ shows that the compound statement is true only when both $P$ and $Q$ are true
  • The truth table for $P \lor Q$ demonstrates that the compound statement is true when at least one of $P$ or $Q$ is true

Proof Techniques

Direct Proof, Proof by Contradiction, and Proof by Induction

• Direct proof starts with given assumptions and proceeds through logical steps to reach the desired conclusion
  • For instance, to prove that the sum of two even integers is always even, start with the assumption that $a$ and $b$ are even integers and show that $a+b$ is also even
• Proof by contradiction assumes the negation of the statement to be proved and derives a contradiction, establishing the truth of the original statement
  • To prove that $\sqrt{2}$ is irrational, assume it is rational and derive a contradiction, thereby proving the original statement
• Proof by induction proves statements involving natural numbers by showing that the statement holds for the base case and that if it holds for an arbitrary natural number, it also holds for the next natural number
  • Mathematical induction can be used to prove that the sum of the first $n$ positive integers is given by $\frac{n(n+1)}{2}$

Other Proof Techniques and Choosing the Appropriate Technique

• Proof by cases proves a statement by considering all possible cases
  • For example, to prove that the product of two integers is even if and only if at least one of the integers is even, consider the cases where both integers are even, both are odd, or one is even and the other is odd
• Proof by counterexample disproves a statement using a single counterexample
  • The statement "all prime numbers are odd" can be disproved by the counterexample of the number 2, which is prime but even
• Choosing the appropriate proof technique depends on the nature of the statement being proved and the available information
  • Direct proof is often used when the conclusion follows straightforwardly from the assumptions
  • Proof by contradiction is useful when the negation of the statement leads to a more easily obtainable contradiction
  • Proof by induction is appropriate for statements involving natural numbers or recursive definitions

Analyzing Mathematical Arguments

Validity and Logical Fallacies

• A valid argument is one in which the conclusion necessarily follows from the premises, assuming the premises are true
  • For example, the argument "All humans are mortal. Socrates is a human. Therefore, Socrates is mortal." is valid because the conclusion follows logically from the premises
• Logical fallacies, such as affirming the consequent, denying the antecedent, and circular reasoning, can lead to invalid arguments
  • Affirming the consequent: "If it rains, the ground gets wet. The ground is wet. Therefore, it rained." (Invalid, as the ground could be wet for other reasons)
  • Denying the antecedent: "If it rains, the ground gets wet. It did not rain. Therefore, the ground is not wet." (Invalid, as the ground could be wet for other reasons)
  • Circular reasoning: "God exists because the Bible says so, and the Bible is true because it is the word of God." (Invalid, as the argument assumes what it is trying to prove)

Assessing Validity and Using Counterexamples

• The validity of an argument can be assessed by examining the logical structure and ensuring that the premises provide sufficient support for the conclusion
  • Valid argument forms, such as modus ponens ($P \Rightarrow Q, P \vdash Q$) and modus tollens ($P \Rightarrow Q, \neg Q \vdash \neg P$), can be used to determine the validity of arguments
• Counterexamples demonstrate the invalidity of an argument by showing a case where the premises are true, but the conclusion is false
  • The argument "All birds can fly. Penguins are birds. Therefore, penguins can fly." is invalid, as the counterexample of penguins, which are birds that cannot fly, disproves the conclusion
• Analyzing the validity of arguments helps identify flaws in reasoning and ensures that mathematical proofs and justifications are sound
  • Identifying invalid arguments prevents the acceptance of false conclusions
  • Recognizing valid argument forms strengthens the foundations of mathematical proofs

Constructing Mathematical Proofs

Structure and Components of a Proof

• A mathematical proof is a logical argument that establishes the truth of a statement beyond any doubt, assuming the axioms and previously proven statements are true
• The structure of a proof typically includes:
  • Stating the proposition to be proved
  • Listing the given information or assumptions
  • Presenting a sequence of logical steps
  • Concluding with the statement that has been proved
• For example, a proof of the statement "The sum of two odd integers is always even" might have the following structure:
  • Proposition: Let $a$ and $b$ be odd integers. Then, $a+b$ is even.
  • Given: $a$ and $b$ are odd integers.
  • Proof:
    1. Since $a$ and $b$ are odd, there exist integers $k$ and $l$ such that $a=2k+1$ and $b=2l+1$.
    2. $a+b = (2k+1) + (2l+1) = 2k + 2l + 2 = 2(k+l+1)$.
    3. Since $k+l+1$ is an integer, $a+b$ is even by the definition of even integers.
  • Therefore, the sum of two odd integers is always even.

Justifications and Common Techniques

• Justifications are explanations or reasons provided for each step in a proof, demonstrating how the step follows logically from the previous steps or known facts
• Common justification techniques include:
  • Citing definitions, axioms, or previously proved theorems
  • Applying inference rules
  • Providing mathematical computations or manipulations
• For instance, in the proof example above, the justifications are:
  • Step 1 uses the definition of odd integers.
  • Step 2 applies algebraic manipulations to rewrite the sum.
  • Step 3 uses the definition of even integers to conclude that $a+b$ is even.
• Constructing clear, concise, and logically sound proofs and justifications is essential for communicating mathematical ideas and establishing the validity of results
  • Well-constructed proofs enable others to understand and verify the reasoning behind mathematical statements
  • Clear justifications make the logical flow of the proof easier to follow and reduce the likelihood of errors or gaps in reasoning