Department

Computer ScienceCourse Code

CSC165H1Professor

Tom FairgrieveThis

**preview**shows half of the first page. to view the full**3 pages of the document.**CSC165H1 Notes

Jan 25, 2012

retain:

P(x) ⇒Q(x)

equivalent to ¬P(x) v Q(x)

¬(∃x∈D, P(x))

⇔

∀x∈D, ¬P(x)

words;

There is no x for which P(x)is true.

equivalent to;

For all x, P(x) is false (¬P(x) is true)

when is

∀x∈D, P(x)

false?

There must be a counter example to P(x) (being true)

∃x∈D, P(x) is false

or

∃x∈D, ¬P(x)

¬(∀x∈D, P(x))

⇔

∃x∈D, ¬P(x)

Negate

All employees earning over 110,000 are female.

Not all employees earning over 110,000 are female

or

There is an employee earning over 110,000 who is not female.

deﬁne O(x): ”x earns over 110,000”

∀x∈O, F(x)

∀x∈E, O(x) ⇒F(x) ←preferred statement

¬(∀x∈E, O(x) ⇒F(x)) ←equiv to. ∃x∈E, ¬(O(x) ⇒F(x))

(O(x) ⇒F(x)) is false when O(x) is True,

but F(x) is false. →O(x) ∧ ¬ F(x)

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