Applied Functional Analysis 3rd Edition (2018) Solutions Manual by J.N. Reddy & J.T. Oden | Complete Chapter Solutions | Functional Analysis Study Guide

,SOLUTIONS MANUAL FOR
Applied Functional
Analysis
THIRD EDITION


by


J. Tinsley Oden
Leszek F. Demkowicz

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1
Preliminaries




Elementary Logic and Set Theory



1.1 Sets and Preliminary Notations, Number Sets



Exercises

Exercise 1.1.1 I f ZI = {. .., −2, −1, 0, 1, 2,.. .} denotes the set of all integers and IN = {1, 2, 3,. . .} the set
of all natural numbers, exhibit the following sets in the form A = {a, b, c, . . .}:

(i) {x ∈ Z
I : x2 − 2x + 1 = 0}
(ii) {x ∈ Z
I : 4 ≤ x ≤ 10}
(iii) {x ∈ IN : x2 < 10}

(i) {1}
(ii) {4, 5, 6, 7, 8, 9, 10}
(iii) {1, 2, 3}




1.2 Level One Logic



Exercises

Exercise 1.2.1 Construct the truth table for De Morgan’s Law:

~ (p ∧ q) ⇔ ((∼ p) ∨ (∼ q))


1

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2 APPLIED FUNCTIONAL ANALYSIS SOLUTION MANUAL




∼ (p ∧ q) ⇔ ((∼ p) ∨ (∼ q))
1 000 1 10 1 10
1 001 1 10 1 01
1 100 1 01 1 10
0 111 1 01 0 01


Exercise 1.2.2 Construct truth tables to prove the following tautologies:
(p ⇒ q) ⇔ (∼ q ⇒∼ p)

∼ (p ⇒ q) ⇔ p ∧ ∼ q



(p ⇒ q) ⇔ (∼ q ⇒ ∼ p)
0 1 0 1 10 1 10
0 1 1 1 01 1 10
1 0 0 1 10 0 01
1 1 1 1 01 1 01
∼ (p ⇒ q) ⇔ p ∧ ∼ q
0 (0 1 0) 1 0 0 1 0
0 (0 1 1) 1 0 0 0 1
1 (1 0 0) 1 1 1 1 0
0 (1 1 1) 1 1 0 0 1
Exercise 1.2.3 Construct truth tables to prove the associative laws in logic:
p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r

p ∧ (q ∧ r) ⇔ (p ∧ q) ∧ r

p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r
00 000 1 000 00
01 011 1 000 11
01 110 1 011 10
01 111 1 011 11
11 000 1 110 10
11 011 1 110 11
11 110 1 111 10
11 111 1 111 11
p ∧ (q ∧ r) ⇔ (p ∧ q) ∧ r
00 000 1 000 00
00 001 1 000 01
00 100 1 001 00
00 111 1 001 01
10 000 1 100 00
10 001 1 100 01
10 100 1 111 00
11 111 1 111 11