Question

(1) Let Q(x, y) be the statement "x+y=xy." If the domain for both variables consists of all integers, what are the truth values?
(a)Q(1,1)
(b)Q(2,0)
(c)∀yQ(1, y)
(d)∃xQ(x,2)
(e)∃x∃yQ(x, y)
(f)∀x∃yQ(x, y)
(g)∃y∀xQ(x, y)
(h)∀y∃xQ(x, y)
(i)∀x∀yQ(x, y)

Answer 1

a. Q(1,1)

$$x+y=1+1=2$$ $$xy=(1) (1)=1$$

Since x+y is not equal to xy, truth value of Q(x, y) is FALSE

Answer: FALSE

b. Q(2,0)

$$x+y=2+0=2$$ $$xy=2*0=0$$

Since x+y is not equal to xy, the truth value of Q(x, y) is FALSE

Answer: FALSE

c. $\forall yQ(1,y)$

By contradiction;

let y be 1:

$$x+y=1+1=2$$ $$xy=1*1=1$$

Since x+y is not equal to xy, the truth value of Q(x, y) is FALSE

Answer: FALSE

d. $\exist xQ(x, 2)$

let x be 2:

$$x+y=2+2=4$$ $$xy=2*2=4$$

Since x+y=xy, the truth value of Q(x, y) is TRUE

Answer : TRUE

e. $\exist x\exists yQ(x, y)$

let x=2 and y=2

$$x+y=2+2=4$$ $$xy=2*2=4$$

Since x+y=xy, the truth value of Q(x, y) is TRUE

Answer : TRUE

f. $\forall x \exists yQ(x, y)$

Let x=1 and y=1:

$$x+y=1+1=2$$ $$xy=1*1=1$$

Since x+y is not equal to xy, then the truth value of Q(x, y) is FALSE

Answer : FALSE

g. $\exist y \forall xQ(x, y)$

Let x=1 and y=1:

$$x+y=1+1=2$$ $$xy=1* 1=1$$

Since x+y is not equal to xy, then the truth value of Q(x, y) is FALSE

Answer : FALSE

h. $\forall y \exist xQ(x, y)$

Let x=1, y=1:

$$x+y=1+1=2$$ $$xy=1* 1=1$$

Since x+y is not equal to xy, then the truth value of Q(x, y) is FALSE

Answer : FALSE

i. $\forall x\forall yQ(x, y)$

Let x=1, y=1:

$$x+y=1+1=2$$ $$xy=1*1=1$$

Since x+y is not equal to xy, then the truth value of Q(x, y) is FALSE

Answer : FALSE