Welcome to Discrete Math
The first rule of Discrete Math
How you will feel at first
“The fool looks at the finger that points to the sky.”
You are learning a new language
You are jumping into a new culture
Can be assigned a truth value (true or false)
Examples of Statements
- Every integer is a perfect square
- \(x\) is an even number
- If it snows I will wear boots
Examples of Non Statements
- What is your favorite number?
- Suppose that \(x\) is even.
Exploring the truth value of a statement
Dip our toes in the water. Convince your partner that:
\(x + 1\) is odd for every even integer \(x\).
To really demonstrate the truth of something we need a proof.
Take a look at my friend's blog
Today let's start with the atoms of proofs
Take a statement \(p\) and a statement \(q\).
We can combine them to create a new statement.
Different combinations have different interpretations
\(p \land q\) is "\(p\) and \(q\)"
\(p \lor q\) is "\(p\) or \(q\)" (inclusive or)
\(p \to q\) is "If \(p\) then \(q\)"
\(\lnot p\) is "Not \(p\)"
\(\exists x\) is "There exists an \(x\)"
\(\forall x\) is "For all \(x\)"
\( p \iff q\) is "\(p\) if and only if \(q\)"
AND and OR
Combine the statement \(p: x+2 = 3\) with \(q: x \neq 1\) using \(\land\) and \(\lor\)
Which new statement is true?