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# CS 064 Week 2 Notes CS 064

UVM

GPA 3.8

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## About this Document

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## Popular in Discrete Structures

## Popular in ComputerScienence

This 3 page Class Notes was uploaded by Lindsay Ross on Wednesday February 3, 2016. The Class Notes belongs to CS 064 at University of Vermont taught by in Winter 2016. Since its upload, it has received 11 views. For similar materials see Discrete Structures in ComputerScienence at University of Vermont.

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Date Created: 02/03/16

If P then Q and If Q then P p q If P then Q and If Q then P T F F nn I I l39I39I l F T The following expressions are equivalent If P then Q P implies Q Q is implied by P P is suf cient for Q Q is necessary for P P III Q QlIlP The following expressions are equivalent 0 P if and only if Q o P iff Q o P is equivalent to Q o P is necessary and suf cient for Q 0 P le Proof Template 1 Direct Proof of lfthen theorem 1 Write the rst sentences of the proof by restating the premise hypothesis of the theorem Choose some appropriate notation 2 Write the last sentence of the proof by restating the conclusion at the bottom of the page 3 Connect these two statements with a logical claim If PZQ is false then the theorem is false if we come up with not Q then the theorem has been disproved Theorem let a denote an even integer and b denote an odd integer Then ab is even 0000000 If a is even and b is odd then ab is even Proof 1 a is even and b is odd Given 2 a is even De nition of and 3 2a De nition of even 4 a2n where n is an element De nition of divide of Z De nition of and 5 b is odd De nition of odd 6 b2m1 where m is an Substitution element of Z Closure of integer multiplication 7ab2n2m1 8 ab2c cn2m1 where c is De nition of divisible an element of Z De nition of even 92ab 10 ab is even Proof Template 1 Premise at the top of the page P conclusion at the bottom of the page Q Work down from P or up from Q such that 2 ends meet and we ve constructed a continuous logical chain of implications that begin at P and end at Q If ab and bc then ac Proof 1 ab and bc Given 2 ab De nition of and 3 bc De nition of and 4 bax where x is an element De nition of divisible of Z De nition of divisible 5 cby where y is an element of Substitution and associative Z property 6 caxyaxy where xy is an element of Z De nition of divisible 7ac Proof template 2 Direct proof of P iff Q 1 Use Proof Template 1 to show how PDQ 2 Use Proof Template 2 to show how QP Proposition Let a and b be an element of Z ab and ba if and only if ab Proof Part 1 1 ab and ba Given 2 axb and by a where x and De nition of divisible y are elements of Z 3 axya Distribution 4 xy1 Cancellation 5 x1 and y1 Law of multiplication 6ab De nition of Part 2 1ab Given 2 a1b Identity of multiplication 3 ab De nition of divisible 4 ab1 Identity of multiplication 5 ba De nition of divisible 6 ab and ba De nition of and

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