Mathematics for Computing

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Mathematics for Computing
Lecture 2 Computer Logic and Truth Tables Dr
Andrew Purkiss-Trew Cancer Research
UK a.purkiss_at_mail.cryst.bbk.ac.uk
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Logic

• Propositions
• Connective Symbols / Logic gates
• Truth Tables
• Logic Laws

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Propositions

• Definition A proposition is a statement that is
  either true or false. Which ever of these (true
  or false) is the case is called the truth value
  of the proposition.

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Connectives

• Compound propositione.g. If Brian and Angela
  are not both happy, then either Brian is not
  happy or Angela is not happy
• Atomic propositionBrian is happy Angela is
  happy
• Connectivesand, or, not, if-then

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Connective Symbols
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Conjugation

• Logical and
• Symbol ?
• Written p ? q
• Alternative forms p q, p . q, pq
• Logic gate version

p
pq
q
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Disjunction

• Logical or
• Symbol ?
• Written p ? q
• Alternative form p q
• Logic gate version

p
p q
q
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Negation

• Logical not
• Symbol
• Written p
• Alternative forms p, p, p
• Logic gate version

p
p
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Truth Tables
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Compound Propositions
(p ? q)
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Tautologies

• Always true

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Contradictions

• Always false

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Website for Lecture Notes

• http//www.cryst.bbk.ac.uk/bpurk01/MfC/index2007.
  html

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End of First Logic 1?

• Place marker

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Mathematics for Computing
Lecture 3 Computer Logic and Truth Tables 2 Dr
Andrew Purkiss-Trew Cancer Research
UK a.purkiss_at_mail.cryst.bbk.ac.uk
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Logical Equivalence

• Logical equals
• Symbol
• Written p p

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Conditional

• Logical if-then
• Symbol ?
• Written p ? q

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Biconditional

• Logical if and only if
• Symbol ?
• Written p ? q

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converse and contrapositive

• The converse of p ? q is q ? p
• The contrapositive of p ? q is q ? p

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Laws of Logic

• Laws of logic allow us to combine connectives and
  simplify propositions and prove that logical
  equivalences are correct.

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Double Negative Law

• p p

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Implication Law

• p ? q p ? q

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Equivalence Law

• p ? q (p ? q) ? (q ? p)

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Idempotent Laws

• p ? p p
• p ? p p

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Commutative Laws

• p ? q q ? p
• p ? q q ? p

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Associative Laws

• p ? (q ? r) (p ? q) ? r
• p ? (q ? r) (p ? q) ? r

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Distributive Laws

• p ? (q ? r) (p ? q) ? (p ? r)
• p ? (q ? r) (p ? q) ? (p ? r)

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Identity Laws

• p ? T p
• p ? F p

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Annihilation Laws

• p ? F F
• p ? T T

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Inverse Laws

• p ? p F
• p ? p T

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Absorption Laws

• p ? (p ? q) p
• p ? (p ? q) p

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de Morgans Laws

• (p ? q) p ? q
• (p ? q) p ? q

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End of Logic