solutions MANUAL FOR

solutions MANUAL FOR Applied Functional Analysis

THIRD EDITION

J.TinsleOden Leszek F. Demkowicz by 1 / 4

1 Preliminaries Elementary Logic and Set Theory 1.1 Sets and Preliminary Notations, Number Sets Exercises Exercise 1.1.1IfIZ={...,−2,−1,0,1,2,...}denotes the set of all integers andIN={1,2,3,...}the set of all natural numbers, exhibit the following sets in the formA={a, b, c, . . .}:

(i){x∈IZ:x

2 −2x+1=0}

(ii){x∈IZ:4≤x≤10}

(iii){x∈IN:x

2 <10} (i){1} (ii){4,5,6,7,8,9,10} (iii){1,2,3} 1.2 Level One LogicExercises

Exercise 1.2.1Construct the truth table forDe Morgan’s Law:

∼(p∧q)⇔((∼p)∨(∼q))

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2APPLIED FUNCTIONAL ANALYSISSOLUTION 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 1.2.2Construct truth tables to prove the following tautologies:

(p⇒q)⇔(∼q⇒∼p) ∼(p⇒q)⇔p∧∼q (p⇒q)⇔(∼q⇒∼p)

010 1 101 10

011 1 011 10

100 1 100 01

111 1 011 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 1.2.3Construct 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 3 / 4

Preliminaries3 1.3 Algebra of Sets Exercises Exercise 1.3.1Of 100 students polled at a certain university, 40 were enrolled in an engineering course, 50 in a mathematics course, and 64 in a physics course. Of these, only 3 were enrolled in all three subjects, 10 were enrolled only in mathematics and engineering, 35 were enrolled only in physics and mathematics, and 18 were enrolled only in engineering and physics.(i) How many students were enrolled only in mathematics?(ii) How many of the students were not enrolled in any of these three subjects?LetA, B, Cdenote the subsets of students enrolled in mathematics, the engineering course and physics, repectively. Sets:A∩B∩C, A∩B−(A∩B∩C),A∩C−(A∩B∩C)andA−(B∪C)are pairwise disjoint (no two sets have a nonempty common part) and their union equals setA, see Fig. 1.1.Consequently,

#(A−(B∪C)) = #A −#A∩B∩C−#(A∩B−(A∩B∩C))−#(A∩C−(A∩B∩C))

= 50−3−10−35=2

In the same way we compute, #(B−(A∪C)) = 9and#(C−(A∪B)) = 8 Thus, the total number of students enrolled is

#(A−(B∪C)) + #(B −(A∪C)) + #(C −(A∪B))

+#(A ∩B−C) + #(A ∩C−B) + #(B ∩C−A)

+#(A ∩B∩C)

=2+9+8+10+35+18+3=85

Consequently, 15 students did not enroll in any of the three classes.Exercise 1.3.2List all of the subsets ofA={1,2,3,4}. Note:Aand∅are considered to be subsets ofA.∅, {1},

{2},{1,2},

{3},{1,3},{2,3},{1,2,3},

{4},{1,4},{2,4},{1,2,4},{3,4},{1,3,4},{2,3,4},{1,2,3,4}

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