Answer to Question #320757 in Discrete Mathematics for Ron

Question

1. a. Construct a truth table for (p ↔ q) and (p → q) ^ (q → p).

b. Determine whether these compound propositions are logically equivalent.

2. Let R(x, y): x² + y² = 1. Find the truth values of the propositions R(2 3 , 6 4 ) and R(3 5 , 7 4 ) .

3. Let P(x, y, z): x + y = z, where x, y and z are all real numbers.

a. Express the quantifications ∀x∀y∃z P(x, y, z) and ∃z∀x∀yP(x, y, z) as statements.

b. Find the truth value of the quantifications ∀x∀y∃z P(x, y, z) and ∃z∀x∀yP(x, y, z).

c. Determine whether both quantifications are logically equivalent.

Answer 1

$2:\\R\left( 23,64 \right) =False:23^2+64^2\ne 1\\R\left( 35,74 \right) =False:35^2+74^2\ne 1\\3:\\a:\\\forall x\forall y\exists z\,\,P\left( x,y,z \right) : for\,\,all\,\,real\,\,x\,\,and\,\,all\,\,real\,\,y\,\,there\,\,exists\,\,real\,\,z\,\,such\,\,that\,\,x+y=z\\\exists z\forall x\forall y: there\,\,exists\,\,such\,\,real\,\,z\,\,that\,\,for\,\,all\,\,x\,\,and\,\,for\,\,all\,\,y\,\,x+y=z\\b:\\\forall x\forall y\exists z\,\,P\left( x,y,z \right) -True, z=x+y\,\,is\,\,real\\\exists z\forall x\forall y\,\,-\,\,False, no\,\,z\,\,can\,\,be\,\,equal\,\,to\,\,all\,\,sums\,\,of\,\,x\,\,and\,\,y\\c:\\Not\,\,equivalent, \sin ce\,\,their\,\,truth\,\,values\,\,are\,\,different$