COMPLETE TEST BANK For Applied Functional Analysis (3rd Edition) By J. Tinsley Oden (Author) Graded A+ Latest Update.

COMPLETE TEST BANK
For Applied Functional Analysis (3rd Edition)
By J. Tinsley Oden (Author)
Graded A+ Latest Update.

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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 Iz = {. .., −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 ∈ Iz : X2 − 2x +1 = 0}
(ii) {X ∈ Iz : 4 ≤ X ≤ 10}

(iii) {X ∈ In : X2 < 10}

(i) {1}

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

, Preliminaries 5
1.2 Level One Logic



Exercises

Exercise 1.2.1 Construct The Truth Table For De Morgan’s Law:

~ (P ∧ Q) ⇔ ((∼ P) ∨ (∼ Q))


1




(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 00 1 0
0 (0 1 1) 1 00 0 1
1 (1 0 0) 1 11 1 0
0 (1 1 1) 1 10 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

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P (Q R) (P Q) R