1. Chapter 1, Section 1 TRUE FALSE F T T F T T F T T T T T T F T T T F
F F T
F T T
negations
ops in ( )
final
𝐶∨𝑆
𝑆 ′ ∨𝐶 ′
𝐶 ′ ∧ 𝑆 ′
𝐶 ′ →𝑆
𝐵→𝑅 ≡ 𝐵 ′ ∨𝑅
𝐵→𝑅
𝐵∧ 𝑅 ′
𝐵 ′ ∧ 𝑅 ′
𝐵′∨𝑅′
𝐵→𝑅 ′ ≡ 𝐵 ′ ∨𝑅 ′ ≡𝐵∧𝑅′
𝐵→𝐶
𝐵→𝐶 ′ ≡ 𝐵 ′ ∨𝐶 ′ ≡𝐵∧𝐶′
𝐵→𝐶′
𝐵′∧𝐶
𝐵∧𝐶′

2. note: necessary ≠ sufficient If A is true then B must be true 𝐴→𝐵
𝐴∨𝐵 →𝐶
𝐶∧ 𝐴∨𝐵 ′
𝐶∧𝐴′∧𝐵′
(𝐴→𝐵)∧( 𝐵 ′ →𝐶)
𝐵→𝐴
note: necessary ≠ sufficient
If A is true then B must be true
𝐴→𝐵
𝑃→(𝐶∧𝐵)
P = project finished
C = client happy
B = bills paid
L = lights out
𝐵′→𝐿
𝑃↔𝐿′
( 𝐵 ′ ∧𝐿)→𝐶′
(𝐵↔𝑃)′→𝐿
𝐵↔(𝑃∨𝐿)

3. A B B->A A->(B->A)
tautology
A B A^B B’ A’ B’v A’ *
contradiction
A B AvB’ A^B (A^B)’ *
A B C AvB (AvB)^C’ A’vC *
A B C A’ BvC’ *

4. Chapter 1 Section 3
TRUE
TRUE
FALSE
TRUE
A(x) is “x=0” B(x,y) is “x times y = 0”
A(x) is “x≠0” B(x,y) is “x times y = 0”
if A(x) is always true then B(x) is always true
for each x for which A(x) then also B(x)
X A(x) B(x)
: : :
A(x) is “x is a surgeon” B(x) is “x is a physician”
A(x) is “x = 42” B(x) is “x <> 42”
P(x) is “x is an integer” Q(x) is “x is a real number”
P(x) is “x is a real number” Q( x) is “x is an integer”
∀𝑥[𝐿 𝑥 →𝑃 𝑥 ]
∃𝑥[𝐿 𝑥 ∧𝑅 𝑥 ]
∀𝑥[𝐿 𝑥 →𝑅 𝑥 ]
∃𝑥[𝐿(𝑥)∧∀𝑦 𝑍 𝑦 →𝐸 𝑥,𝑦 ]
∀𝑥∀𝑦[ 𝐿 𝑥 ∧𝑍 𝑦 →𝐸 𝑥,𝑦 ]

5. ∀𝑥(¬𝑃 𝑥 )
∃𝑥(𝑃 𝑥 )
∀𝑥 𝐶 𝑥 ∧𝐿 𝑥
∃𝑥 𝐶′ 𝑥 ∨𝐿′ 𝑥
((∃𝑥 𝐴 𝑥 ′ ′ ≡∀(𝑥)[𝐴 𝑥 ]′′≡∀(𝑥)(𝐴 𝑥 )
Let P(x) be false for all x
Let Q(x) be true for some x
∀𝑥(𝑃 𝑥 →𝑄 𝑥 )
Will always be true (vacuously)
∃𝑥(𝑃 𝑥 ∨𝑄 𝑥 )≡∃𝑥(𝑄 𝑥 )
Since P(x) is never true
But then
∃𝑥(𝑃 𝑥 ∧𝑄 𝑥 )
Can never be true
Let x be the domain of dogs
Let y be the domain of cats
Let P(x) be “x purrs”
Let Q(x) be “x barks”
In English: If all dogs purr or bark then either all dogs purr or there exists a cat that barks.

6. Are these system specifications consistent?
The router can send packets to the edge system only if it supports the new address space.
For the router to support the new address space it is necessary that the latest software release be installed.
The router can send packets to the edge system if the latest software release is installed.
The router does not support the new address space
R -> S
S -> I
I -> R ~S
R = router can send packets to the edge system
S = router support new address space
I = latest SW release is installed
R S I R->S S->I I->R ~S
system is consistent