Logical Negation

• a proposition is a statement that is either true or false. Examples:
  • if $n$ is an even integer, then $n^2$ is an even integer. (happens to be true)
  • $\empty \in \mathbb R$. (happens to be false)
• the negation of a proposition $X$ is a proposition that is true whenever $X$ is false and is false whenever $X$ is true.
  • e.g. for the proposition “it is snowing outside,” its negation is “it is not snowing outside.”

How to find the negation of a statement:

• e.g. “All My Friends Are Taller Than Me” (universal statement)
  • negation: something like “there exists a friend that is not taller than me.” (existential statement:
  • pattern: the negation of “For all $x$, $P(x)$ is true” (universal) is “There exists an $x$ that such that $P(x)$ is false” (existential)
• e.g. “Some Friend is Shorter Than Me” (existential statement)
  • negation: something like “all my friends are taller than me.”
  • pattern: the negation of “There exists an $x$ where $P(x)$ is true” (existential) is “For all $x$, $P(x)$ is false” (universal)
• under negation, universals become existentials and existentials become universals
  • other examples: “every brown dog loves every orange cat” → “there exists a brown dog that does not love some orange cat.”
    • “there exists a brown dog that does not love every orange cat” is a lot more problematic; it could be used to mean that a brown dog hates every orange cat OR a brown dog has an orange cat that it does not love.

Proof by Contradiction

• every statement in mathematics is either true or false. If statement $P$ is not false, it means $P$ must be true.

• A proof by contradiction shows that some statement $P$ is true by showing that $P$ isn’t false.

  • First, assume $P$ is false. The goal is to show that this assumption is silly.
  • Next, show that this leads to an impossible result.
  • Finally, conclude that since $P$ can’t e false, we know that $P$ is true.

• First example (set cardinalities). Is there a “largest” set? Is there a set that is bigger than every other set?

  • Theorem: There is no largest set.
  • Proof: Assume for the sake of contradiction that there is a largest set; call it $S$. Now, consider the set $\wp(S)$. By Cantor’s Theorem, we know that $|S| < |\wp(S)|$, so $\wp(S)$ is a larger set than $S$. This contradicts the fact that $S$ is the largest set. We’ve reached a contradiction, so our assumption must have been wrong. Therefore, there is no largest set. $\blacksquare$

• Notes: announce a proof by contradiction, and show what we’re assuming to be true. In CS 103, we need to include these three pieces

  • say that the proof is by contradiction
  • say what you are assuming is the negation of the statement to prove
  • say you have reached a contradiction and what the contradiction means.

• Another example: a Latin square is an $n \times n$ grid filled with the numbers $1,2,\dots,n$ such that every number appears in each row and each column exactly once. The main diagonal of a Latin square runs from the top-left corner to the bottom-right corner. A Latin Square is symmetric if the numbers are symmetric across the main diagonal.

  • Theorem: Every symmetric Latin square of odd size $n \times n$ has each of the numbers$1,2,\dots,n$ on its main diagonal.
  • Proof: Assume for the sake of contradiction that there is a symmetric Latin square of odd size $n \times n$ that does not have one of the numbers $1,2,\cdots,n$ on its main diagonal. Call the missing number $r$.

  Let $k$ be the number of times $r$ appears above the main diagonal. Since the Latin square is symmetric, there are also $k$ copies of $r$ below the main diagonal. And because $r$ doesn’t appear on the main diagonal, that accounts for all copies of $r$, so there are exactly $2k$ copies of $r$.

  Independently, we know that $r$ appears $n$ times in the Latin square, once for each of its $n$ rows.

  Combining these results, we see that $n=2k$. This means that $n$ is even, contradicting the fact that $n$ is odd. We’ve reached a contradiction, so our assumption is wrong. Therefore, all symmetric Latin squares of odd size $n\times n$ have each of the numbers $1,2,\dots,n$ on their main diagonals. $\blacksquare$

Logical Implication

Implication: If $P$ is true, then $Q$ is true.

• the first part of the implication (the if part) is called the antecedent
• the second part of the implication (the then part) is called the consequent
• implications are directional. The implication above does not imply if $Q$ is true, then $P$ is true.
• Implications only say something about the consequent when the antecedent is true. If $P$ is false, that doesn’t mean $Q$ is false.
• No discussion of causation.