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Day 56 – Truth Tables

Table of contents

Review: Each statement is TRUE (1) or FALSE (0).

  • 32 is even. T or 1
  • A ⊆ B iff x ∈ B implies x ∈ A F or 0

All connectives take a truth value and output a truth value.It means that depending on the connective, the truth value can change.

Negation

The negation is always the opposite of the truth value. If p is true (1), then the negation is false (0).

Mathematical Value

¬P  =  1  −  PP  =  1therefore,  ¬P  =  1−P  =0\neg P\;=\;1\;-\;P\\P\;=\;1\\therefore,\;\neg P\;=\;1-P\;=0

Conjunction (and)

In conjunction we will see ∧ (caret), & (and symbol), or . (dot).

Conjunction takes two statements and combine them.

#  of  rows  =  2number  of  statements\#\;of\;rows\;=\;2^{number\;of\;statements}

Mathematical Value

p  ∧  q  =  min(p,q)p\;\wedge\;q\;=\;min(p,q)

Disjunction (or)

In disjunction we will see ∨ or +.

Takes either or both conditions of p or q.

Mathematical Value

p  ∧  q  =  max(p,q)p\;\wedge\;q\;=\;max(p,q)

Conditional (if, then)

We would normally see → or ⊃ symbols.

Mathematical Value

p  →  q  =  1iffp  ≤  qp\;\to\;q\;=\;1\\iff\\p\;\leq\;q

Biconditional (if and only if, iff)

Symbols used: ↔ ⇔ ≡ ⟺

Mathematical Value

p  =  q  then  p    ⟺    q  =  1p\;=\;q\;then\;p\;\iff\;q\;=\;1

Exclusive Or (the idea of or in English)

Symbols used: ⊻ or ⊕

pqp ⊕ q
110
101
011
000
p≠  q  then  p  ⊕  q  =1p \neq\;q\;then\;p\;\oplus\;q\;=1

Latex:

  • subset of = ⊆ = \subseteq
  • not subset of = ⊈ = \nsubseteq
  • element of = ∈ = \in
  • not element of = ∉ = \notin
  • and = ∧ = \wedge
  • or = ∨ = \vee
  • if and only if = ↔ = \iff
  • exclusive or = ⊕ = \oplus

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