Answer to Question #132726 in Discrete Mathematics for Promise Omiponle

Question #132726
(1) Let Q(x, y) be the statement "x+y=x-y." 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)
1
Expert's answer
2020-09-23T17:38:24-0400

Solution to a:

Q(1,1)


x+y=1+1=2x+y=1+1=2xy=11=0x-y=1-1=0

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

Ans: FALSE


Solution to b:

Q(2,0)


x+y=2+0=2x+y=2+0=2xy=20=2x-y=2-0=2

since x+y=x-y, the truth value of Q(x,y) is TRUE.

Ans: TRUE


Solution to c:

yQ(1,y)\forall y Q(1,y)

let y=1;


x+y=1+1=2x+y=1+1=2xy=11=0x-y=1-1=0

since y\exists y for which x+y is not equal to x-y (e.g y=1), the truth value of Q(x,y) is FALSE.

Ans: FALSE


Solution to d:

xQ(x,2)\exists xQ(x,2)


x+2=x2x+2=x-2xx=22x-x=-2-20=40=-4

since the above equation does not give a definite solution of x, the truth value of Q(x,y) is FALSE.

Ans: FALSE


Solution to e:

xyQ(x,y)\exists x\exists yQ(x,y)

let x=2, y=0;


x+y=2+0=2x+y=2+0=2xy=20=2x-y=2-0=2

since x\exists x and y\exists y such that x+y=x-y (i.e x=2, y=0), the truth value of Q(x,y) is TRUE.

Ans: TRUE


Solution to f:

xyQ(x,y)\forall x \exists y Q(x,y)

let x=1,y=1;


x+y=1+1=2x+y=1+1=2xy=11=0x-y=1-1=0

since x+y is not equal x-y when x=1 and y=1, the truth value of Q(x,y) is FALSE

Ans: FALSE


Solution to g:

yxQ(x,y)\exists y \forall xQ(x,y)

when y=0;

x+0=x0x+0=x-0x=xx=x

since x+y=x-y for all x when y=0, the truth value of Q(x,y) is TRUE

Ans: TRUE


Solution to h:

yxQ(x,y)\forall y \exists x Q(x,y)

let y=1;


x+1=x1x+1=x-1xx=11x-x=-1-10=20=-2

since the above equation does not give a definite solution of x, the truth value of Q(x,y) is FALSE.

Ans: FALSE


Solution to i:

xyQ(x,y)\forall x\forall yQ(x,y)

let x=1, y=1;


x+y=1+1=2x+y=1+1=2xy=11=0x-y=1-1=0

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

Ans: FALSE



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