In each of the following ,two open statements P(x,y) and Q(x,y) are given,where the domain

Question

of both x and y is z. determine the truth value of P(x,y) => Q(x,y) for the given values of x and y
with explanation
a) P(x,y):x2 - y2=0 and Q(x,y)ix=y.(x,y)ef(1,-1),(3,4),(5,5))
ON
P(x.y):|x|=ly and Q(x.,y):x = y.(x.y)ef(1,2),(2,-2),(6,6))
0) Plxy)ux2 + y2= 1 and Q(xyix +y=1.(x,y)e((1,-1),(3,4),(0,-1),(1,0))

Answer 1

To determine the truth value of the statement \( P(x,y) \Rightarrow Q(x,y) \) for the given values of \( (x, y) \), we need to evaluate both \( P(x, y) \) and \( Q(x, y) \) independently for each pair, and then combine the results using the logical implication.

The implication \( P(x,y) \Rightarrow Q(x,y) \) is true unless \( P(x,y) \) is true and \( Q(x,y) \) is false. The truth table for implication \( A \Rightarrow B \) is:

• True when both \( A \) is true and \( B \) is true.
• True when \( A \) is false (regardless of \( B \)).
• False when \( A \) is true and \( B \) is false.

Let's evaluate each case one by one.

Part (a)

Given:

• \( P(x,y): x^2 - y^2 = 0 \)
• \( Q(x,y): x = y \)
• Pairs: \( (1,-1), (3,4), (5,5) \)

1. For \( (1,-1) \):

   • \( P(1, -1) \): \( 1^2 - (-1)^2 = 1 - 1 = 0 \) (True)
   • \( Q(1, -1) \): \( 1 = -1 \) (False)
   • Truth Value: \( P(1, -1) \Rightarrow Q(1, -1) \) is False (True => False).

2. For \( (3,4) \):

   • \( P(3, 4) \): \( 3^2 - 4^2 = 9 - 16 = -7 \) (False)
   • \( Q(3, 4) \): \( 3 = 4 \) (False)
   • Truth Value: \( P(3, 4) \Rightarrow Q(3, 4) \) is True (False => anything is True).

3. For \( (5,5) \):

   • \( P(5, 5) \): \( 5^2 - 5^2 = 25 - 25 = 0 \) (True)
   • \( Q(5, 5) \): \( 5 = 5 \) (True)
   • Truth Value: \( P(5, 5) \Rightarrow Q(5, 5) \) is True (True => True).

Summary for (a):

• \( (1,-1) \): False
• \( (3,4) \): True
• \( (5,5) \): True

Part (b)

Given:

• \( P(x,y): |x| = |y| \)
• \( Q(x,y): x = y \)
• Pairs: \( (1,2), (2,-2), (6,6) \)

1. For \( (1,2) \):

   • \( P(1, 2) \): \( |1| = |2| \) (1 = 2, False)
   • \( Q(1, 2) \): \( 1 = 2 \) (False)
   • Truth Value: \( P(1, 2) \Rightarrow Q(1, 2) \) is True (False => anything is True).

2. For \( (2,-2) \):

   • \( P(2, -2) \): \( |2| = |-2| \) (2 = 2, True)
   • \( Q(2, -2) \): \( 2 = -2 \) (False)
   • Truth Value: \( P(2, -2) \Rightarrow Q(2, -2) \) is False (True => False).

3. For \( (6,6) \):

   • \( P(6, 6) \): \( |6| = |6| \) (True)
   • \( Q(6, 6) \): \( 6 = 6 \) (True)
   • Truth Value: \( P(6, 6) \Rightarrow Q(6, 6) \) is True (True => True).

Summary for (b):

• \( (1,2) \): True
• \( (2,-2) \): False
• \( (6,6) \): True

Part (c)

Given:

• \( P(x,y): x^2 + y^2 = 1 \)
• \( Q(x,y): x + y = 1 \)
• Pairs: \( (1,-1), (3,4), (0,-1), (1,0) \)

1. For \( (1,-1) \):

   • \( P(1, -1) \): \( 1^2 + (-1)^2 = 1 + 1 = 2 \) (False)
   • \( Q(1, -1) \): \( 1 + (-1) = 0 \) (False)
   • Truth Value: \( P(1, -1) \Rightarrow Q(1, -1) \) is True (False => anything is True).

2. For \( (3,4) \):

   • \( P(3, 4) \): \( 3^2 + 4^2 = 9 + 16 = 25 \) (False)
   • \( Q(3, 4) \): \( 3 + 4 = 7 \) (False)
   • Truth Value: \( P(3, 4) \Rightarrow Q(3, 4) \) is True (False => anything is True).

3. For \( (0,-1) \):

   • \( P(0, -1) \): \( 0^2 + (-1)^2 = 0 + 1 = 1 \) (True)
   • \( Q(0, -1) \): \( 0 + (-1) = -1 \) (False)
   • Truth Value: \( P(0, -1) \Rightarrow Q(0, -1) \) is False (True => False).

4. For \( (1,0) \):

   • \( P(1, 0) \): \( 1^2 + 0^2 = 1 + 0 = 1 \) (True)
   • \( Q(1, 0) \): \( 1 + 0 = 1 \) (True)
   • Truth Value: \( P(1, 0) \Rightarrow Q(1, 0) \) is True (True => True).

Summary for (c):

• \( (1,-1) \): True
• \( (3,4) \): True
• \( (0,-1) \): False
• \( (1,0) \): True

Final Summary of Truth Values

• Part (a): \( (1,-1) \) - False, \( (3,4) \) - True, \( (5,5) \) - True
• Part (b): \( (1,2) \) - True, \( (2,-2) \) - False, \( (6,6) \) - True
• Part (c): \( (1,-1) \) - True, \( (3,4) \) - True, \( (0,-1) \) - False, \( (1,0) \) - True