lect05_s.pdf - Discrete Mathematics for Computing Mat1101...

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Discrete Mathematics for Computing Mat1101 Logic, Laws of Logic, and sets School of Agricultural, Computational and Environmental Sciences Faculty of Health, Engineering and Sciences Lecture 5 Mat1101 1
Lecture Summary 1 Truth tables definitions 2 Inverse, converse and contrapositive 3 Other logical definitions 4 Laws of logic 5 Order of logical operations 6 Introduction to sets Special sets Venn diagrams 7 Predicate logic Definition Quantifiers Negations of quantified statements Mat1101 2
Truth Table: NOT ( ¬ ) ? ¬? T F F T Negation of the original proposition. Reverses the truth value of proposition ? . (i.e. if ? is True then ¬? is False etc.) Mat1101 3
Truth Table: AND ( ) ? ? ? ∧ ? T T T T F F F T F F F F Only true when both propositions are true. “It is hot and sunny” is understood to be true when both conditions are satisfied. Mat1101 4
Truth Table: OR ( ) ? ? ? ∨ ? T T T T F T F T T F F F True when either ? is true OR ? is true, or both ? and ? are true. Inclusive OR statement , as it includes the case when both are true. Will look at the logic for exclusive case (i.e. ? or ? but not both) later. “Cream or sugar” is understood to mean cream or sugar or both. Mat1101 5
Truth Table: Implication ( ) Consider the proposition: “IF you show up for work on Monday, you will get the job.” Under what circumstances can we say the person spoke falsely? You show up for work on Monday, and do not get the job. The statement says nothing if the condition you (“you show up for work on Monday”) is not meet. If ? is true then ? → ? depends on ? . If ? is false, then ? → ? is said to be true ‘by default’ or ‘vacuously’ true. A further discussion on this can be found at: conditional . ? ? ? → ? T T T T F F F T T F F T Mat1101 6
Inverse, converse and contrapositive of an implication Consider the propositions ? ≡ I am happy and ? ≡ my girlfriend is happy. Implication: ? → ? (if ? then ? ) – If I am happy then so is my girlfriend. Inverse: ¬? → ¬? (if not ? then not ? ) – If I am not happy then neither is my girlfriend. Converse: ? → ? ( if ? then ? ) – If my girlfriend is happy then so am I. Contrapositive: ¬? → ¬? (if not ? then not ? ) – If my girlfriend is not happy then neither am I. Mat1101 7
Some notes on Inverse, Contrapositive and Converse The contrapositive is the inverse of converse. The implication ? → ? is logically equivalent to contrapositive ( ¬? → ¬? ). How do we prove this? Mat1101 8
Truth Table: ? → ? ≡ ¬? → ¬? ? ? ? → ? ¬? ¬? ¬? → ¬? T T T F F T T F F T F F F T T F T T F F T T T T The shaded columns are logical equivalent. Mat1101 9
Some more notes on Inverse, Contrapositive and Converse An implication and its converse are not logically equivalent.

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