Problem 1 Consider the sets A = {a, b, c), B = { 1, 2), C = {b, c } . List the e

Question

Problem 1 Consider the sets A = {a, b, c), B = { 1, 2), C = {b, c } . List the elements of the following sets: a) D=(A× B)-(CXB) b) E-2C-2 c) F 2(2c) e) H={(x, y, z) | x, y, zeBAXtyAX=z) Problem 2 Identify the error or errors in this argument that supposedly shows that if 3x P(x) 3x Q(x) is true then 3x (P(x) Q(x)) is true. 1.3x P(x) 3x Q(x) 2. 3x P(x) 3. P(c) 4.3x Q(x) 5. Q(c) 6. P(c) Q(c) 7.3x (P(x) Q(x)) Premise Simplification from (1) Existential instantiation from (2) Simplification from (1) Existential instantiation from (4) Conjunction from (3) and (5) Existential generalization

Explanation / Answer

Problem 1

(a) D = (A x B) – (C x B)

A x B = {(a,1), (a,2), (b,1),(b,2),(c,1),(c,2)}

C x B = {(b,1),(b,2),(c,1),(c,2)}

D = {(a,1), (a,2)}

b) E = 2C – 2A diff of power set of C minus power set of A)

    E = {, b, c , (b,c)} – {, a,b,c,(a,b),(b,c),(a,c), (a,b,c)}

    E = {}

c) F= 2(2C)        Power set of power set of C

     F = 2{, b, c , (b,c)} = {,, b, c , (b,c), (,b),( ,c),( ,(b,c))(b,c),(b,(b,c))(c,(b,c)),( ,b,c),( ,c,(b,c)),( ,b,(b,c)),(b,c, (b,c)), (, b, c , (b,c))}                                  Remove redundant elements

d) G = (A x B x C) (B x C x A)

G = {(a,1,b), (a,2,b), (b,1,b),(b,2,b),(c,1,b),(c,2,b), (a,1,c), (a,2,c), (b,1,c),(b,2,c),(c,1,c), (c,2,c)} {(1,b,a),(1,c,a),(2,b,a),(2,c,a), (1,b,b),(1,c,b),(2,b,b),(2,c,b) (1,b,c),(1,c,c),(2,b,c),(2,c,c)}

G =

e) H = {(1, 2, 1),(2, 1 ,2)}

f) I = {(a,1,b),(a,1,c),(b,1,c),(c,1,b), (a,2,b),(a,2,c),(b,2,c),(c,2,b),}

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