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Oct 25 In-Class Exercise Thread.

Post your solutions to the Oct 25 In-Class Exercise here.

Best,

Chris

Post your solutions to the Oct 25 In-Class Exercise here. Best, Chris

-- Oct 25 In-Class Exercise Thread

Oindril Dutta

Given Model M with universe {a, b}:

V(z) = a

Parent: {

(a, b) => True
(b, a) => False
(b, b) => False
(a, a) => False

}

(Edited: 2017-10-25)

Oindril Dutta ---- Given Model M with universe {a, b}: V(z) = a Parent: { * (a, b) => True * (b, a) => False * (b, b) => False * (a, a) => False }

-- Oct 25 In-Class Exercise Thread

Universe: {Jim, Jane }

Parent: { (Jim, Jane) = true, (Jane, Jim) =false, (Jim, Jim) = false, (Jane, Jane) = false}

V(z) = Jim

Student: Xin Yan

(Edited: 2017-10-25)

Universe: {Jim, Jane } Parent: { (Jim, Jane) = true, (Jane, Jim) =false, (Jim, Jim) = false, (Jane, Jane) = false} V(z) = Jim Student: Xin Yan

-- Oct 25 In-Class Exercise Thread

 
 
M = {set of people P1, P2, P3}
Parent(x,y)
Parent(x,y) returns true if y is a parent of x 
			false if y is not a parent of x
			we assume that a person x can't be a parent of themselves 
			and a person y can't be a parent of themselves

Parent(P1,P1) = False Parent(P1,P2) = False, P2 is not a Parent of P1 Parent(P1, P3) = False P3 is not a parent of P1 Parent(P2,P2) = False Paren(P2,P1) = False, P1 is not a Parent of P2 Parent(P2, P3) = True, P3 is a parent of P3 Parent(P3,P3) = False Parent(P3, P1) = False, P1 is not a Parent of P2 Parent (P3, P2) = True, P3 is a parent of P2 z = P3 Parent(X,X) = F !Parent(X,X) ^ (There Exists Y)Parent(Z,Y) T ^ T = T

<pre> M = {set of people P1, P2, P3} Parent(x,y) Parent(x,y) returns true if y is a parent of x false if y is not a parent of x we assume that a person x can't be a parent of themselves and a person y can't be a parent of themselves Parent(P1,P1) = False Parent(P1,P2) = False, P2 is not a Parent of P1 Parent(P1, P3) = False P3 is not a parent of P1 Parent(P2,P2) = False Paren(P2,P1) = False, P1 is not a Parent of P2 Parent(P2, P3) = True, P3 is a parent of P3 Parent(P3,P3) = False Parent(P3, P1) = False, P1 is not a Parent of P2 Parent (P3, P2) = True, P3 is a parent of P2 z = P3 Parent(X,X) = F !Parent(X,X) ^ (There Exists Y)Parent(Z,Y) T ^ T = T </pre>

-- Oct 25 In-Class Exercise Thread

Let M contain the universe {1, 2, 3} and interpret Parent(x, y) as x is greater than y.

 Parent(1, 1) => False
 Parent(2, 2) => False
 Parent(3, 3) => False
 Parent(1, 2) => False
 Parent(1, 3) => False
 Parent(2, 3) => False
 Parent(3, 1) => True
 Parent(3, 2) => True
 Parent(2, 1) => True

(∀x)¬Parent(x,x) holds as shown above for Parent(1,1), Parent(2,2), and Parent(3,3).

(∃y)Parent(z,y) holds for the assignment z = 3, y = 2 Therefore the whole statement holds.

(Edited: 2017-10-25)

Let M contain the universe {1, 2, 3} and interpret Parent(x, y) as x is greater than y. Parent(1, 1) => False Parent(2, 2) => False Parent(3, 3) => False Parent(1, 2) => False Parent(1, 3) => False Parent(2, 3) => False Parent(3, 1) => True Parent(3, 2) => True Parent(2, 1) => True (∀x)¬Parent(x,x) holds as shown above for Parent(1,1), Parent(2,2), and Parent(3,3). (∃y)Parent(z,y) holds for the assignment z = 3, y = 2 Therefore the whole statement holds.

-- Oct 25 In-Class Exercise Thread

Predicate: Parent(x,y)
Formula: (∀x)¬Parent(x,x)∧(∃y)Parent(z,y)
Universe M = {Wayne, Jonathan}
Parent(x, y) is true if x is a parent of y:
Parent:

 { (Wayne, Jonathan) => true

 (Wayne, Wayne) => false

 (Jonathan, Wayne) => false

 (Jonathan, Jonathan) => false }

V(z) := Wayne

Predicate: Parent(x,y) Formula: (∀x)¬Parent(x,x)∧(∃y)Parent(z,y) Universe M = {Wayne, Jonathan} Parent(x, y) is true if x is a parent of y: Parent: { (Wayne, Jonathan) => true (Wayne, Wayne) => false (Jonathan, Wayne) => false (Jonathan, Jonathan) => false } V(z) := Wayne

-- Oct 25 In-Class Exercise Thread

Universe: {Apple, Peach }
Parent: { (Apple, Peach) = true, (Peach, Apple) =false, (Apple, Apple) = false, (Peach, Peach) = false}
V(z) = Apple

(Edited: 2017-10-25)

Universe: {Apple, Peach } Parent: { (Apple, Peach) = true, (Peach, Apple) =false, (Apple, Apple) = false, (Peach, Peach) = false} V(z) = Apple

-- Oct 25 In-Class Exercise Thread

M={zoe, jon, frank, greg, =, t, f}

we interpret t to be true and f to be false .

We interpret the predicate = as the usual equality of two natural numbers

parent(zoe, jon)=t

parent(frank, jon)=t

parent(jon, greg)=t

parent(frank,frank)=f

parent(frank,jon)=f

parent(frank,zoe)=f

parent(frank,greg)=f

parent(jon, jon)=f

parent(jon,zoe)=f

parent(jon,frank)=f

parent(jon, greg)=f

parent(greg,greg)=f

parent(greg,jon)=f

parent(greg,frank)=f

parent(greg, zoe)=f

parent(zoe, zoe)=f

parent(zoe, frank)=f

parent(zoe, jon)=f

parent(zoe, greg)=f

(Edited: 2017-10-25)

M={zoe, jon, frank, greg, =, t, f} we interpret t to be true and f to be false . We interpret the predicate = as the usual equality of two natural numbers parent(zoe, jon)=t parent(frank, jon)=t parent(jon, greg)=t parent(frank,frank)=f parent(frank,jon)=f parent(frank,zoe)=f parent(frank,greg)=f parent(jon, jon)=f parent(jon,zoe)=f parent(jon,frank)=f parent(jon, greg)=f parent(greg,greg)=f parent(greg,jon)=f parent(greg,frank)=f parent(greg, zoe)=f parent(zoe, zoe)=f parent(zoe, frank)=f parent(zoe, jon)=f parent(zoe, greg)=f

-- Oct 25 In-Class Exercise Thread

 (∀x)¬Parent(x,x)∧(∃y)Parent(z,y)
 In English:
 There exists for all item values(x) that a parent of x is not x and there 
 exists for some item value(y) that a parent of z is y.

 Model = {"Bob", "Sally", "Suzie"}
 Language = {∀, ∃, ¬, ∧, x, y, z, Parent(value1,value2)}
 Where:
 ∀ := there exists for all item values where the following statement is true
 ∃ := there exists for some item values where the following statement is true
 ¬ := logical "negation"
 ∧ := logical "and"
 x := "Bob"
 y := "Sally"
 z := "Suzie"
 Parent(value1,value2) := the value1 is a parent of the value2
 	Where the values of Parent(value1, value2) exist as:
 	Parent {
 		{"Bob", "Bob"} = False; 
                {"Sally", "Sally"} = False; 
                {"Suzie", "Suzie"} = False;
 		{"Bob", "Sally"} = True; {"Suzie", "Sally"} = True; 
 		{"Sally", "Bob"} = False; {"Sally", "Suzie"} = False;
 		{"Suzie", "Bob"} = False; {"Bob", "Suzie"} = False;
 	}
 
 A case in which the statement (∀x)¬Parent(x,x)∧(∃y)Parent(z,y) holds is: 
 
 (∀M xM) ¬M Parent(xM,xM)M ∧M (∃M yM) Parent(zM,yM)M
 (∀"Bob") ¬ Parent("Bob","Bob") ∧ (∃"Sally") Parent("Suzie","Sally")

(Edited: 2017-10-25)

(∀x)¬Parent(x,x)∧(∃y)Parent(z,y) In English: There exists for all item values(x) that a parent of x is not x and there exists for some item value(y) that a parent of z is y. Model = {"Bob", "Sally", "Suzie"} Language = {∀, ∃, ¬, ∧, x, y, z, Parent(value1,value2)} Where: ∀ := there exists for all item values where the following statement is true ∃ := there exists for some item values where the following statement is true ¬ := logical "negation" ∧ := logical "and" x := "Bob" y := "Sally" z := "Suzie" Parent(value1,value2) := the value1 is a parent of the value2 Where the values of Parent(value1, value2) exist as: Parent { {"Bob", "Bob"} = False; {"Sally", "Sally"} = False; {"Suzie", "Suzie"} = False; {"Bob", "Sally"} = True; {"Suzie", "Sally"} = True; {"Sally", "Bob"} = False; {"Sally", "Suzie"} = False; {"Suzie", "Bob"} = False; {"Bob", "Suzie"} = False; } A case in which the statement (∀x)¬Parent(x,x)∧(∃y)Parent(z,y) holds is: (∀M xM) ¬M Parent(xM,xM)M ∧M (∃M yM) Parent(zM,yM)M (∀"Bob") ¬ Parent("Bob","Bob") ∧ (∃"Sally") Parent("Suzie","Sally")

-- Oct 25 In-Class Exercise Thread

So the formula basically says that, for all 'x', x is not a parent of itself and there exists a 'y' s.t 'z' is the parent of 'y'.

Given model M and the universe (Alice, Bob); V(z) = Bob, assigning z to Bob; Parent = [(Alice, Alice) = False; (Alice, Bob) = False; (Bob, Bob) = False; (Bob, Alice) = True]

(Edited: 2017-10-25)

So the formula basically says that, for all 'x', x is not a parent of itself and there exists a 'y' s.t 'z' is the parent of 'y'. Given model M and the universe (Alice, Bob); V(z) = Bob, assigning z to Bob; Parent = [(Alice, Alice) = False; (Alice, Bob) = False; (Bob, Bob) = False; (Bob, Alice) = True]