Let A and B be any given sets. Use set builder notation and logical equivalences to show that AUB=An B.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Problem 1
Let A and B be any given sets. Use set builder notation
and logical equivalences to show that
AUB=AN B.
Transcribed Image Text:Problem 1 Let A and B be any given sets. Use set builder notation and logical equivalences to show that AUB=AN B.
TABLE 6 Logical Equivalences.
Equivalence
PAT P
PVF p
PVT T
PAF F
pvp p
PAP p
pvqqvp
PAq=qAp
(pvq) vr
(PA)Ar
pv (qvr)
p^(q^r)
PV (q Ar)
(pvq)^(pvr)
PA(qVr) (PAq) v (p^r)
p^q)p
(pvq)=pAny
pv (p^q) = p.
PA(pvq) p
PV-P=T
Name
Identity laws
Domination laws
Idempotent laws
Double negation law
Commutative laws
Associative laws
Distributive laws
De Morgan's laws
Absorption laws
Negation laws
Conditional Statements
pqp Vq
Definition of implication
p-qq-p
Contrapositive
pq=(pq) ^ (q→ p) Definition of biconditional
These 2 laws allow us to write multiple conjunctions (or
disjunctions) in any order without parenthesis, i.e.
pvqvr qvpVr PAqAr=rAp^q
We already proved the first De Morgan's law
Transcribed Image Text:TABLE 6 Logical Equivalences. Equivalence PAT P PVF p PVT T PAF F pvp p PAP p pvqqvp PAq=qAp (pvq) vr (PA)Ar pv (qvr) p^(q^r) PV (q Ar) (pvq)^(pvr) PA(qVr) (PAq) v (p^r) p^q)p (pvq)=pAny pv (p^q) = p. PA(pvq) p PV-P=T Name Identity laws Domination laws Idempotent laws Double negation law Commutative laws Associative laws Distributive laws De Morgan's laws Absorption laws Negation laws Conditional Statements pqp Vq Definition of implication p-qq-p Contrapositive pq=(pq) ^ (q→ p) Definition of biconditional These 2 laws allow us to write multiple conjunctions (or disjunctions) in any order without parenthesis, i.e. pvqvr qvpVr PAqAr=rAp^q We already proved the first De Morgan's law
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