Determine whether ∀x(p(x) ↔ q(x)) and ∀x p(x) ↔ ∀xq(x) are logically equivalent. justify your answer.
Answers
solution:
Given: the two expression which are: ∀x (P(x) → Q(x)) and ∀xP(x) → ∀xQ(x)
To prove : the given expression are equivalent
Proof :
For the two expression to be equivalent, fruth value of both the expressions. Should be same. To determine that the expressions are equivalent or not, it is required to determine that for some specific predicates and specific members of the domain, the truth values are same or not.
Let p(x) = x is an even number
Let Q(x) = x is odd number
According to expression ∀x (P(x) every number is an even number, which is false because all numbers cannot be even
According to the expression, ∀x (Q(x), every number is an odd number, which is false because all numbers cannot be odd.
The given expression: ∀x (P(x) → Q(x)) is false because p(x) and Q(x) are opposite values, which implies that ∀x (P(x) → Q(x)) is false for all x.
Consider the expression : ∀x P(x) → Q(x) in which each of the statement ∀x P(x) and ∀x Q(x), both are false. This implies that the propositions P(x) and Q(x) are true.
Answer:
Let x/a = y/b = z/c = k, [By k method]
x = ak, y= bk and z=ck
L.H.S. = a3k3/a2 + b3k3/b2 + c3k3/c2 > k3[a + b + c]
R.H.S. = [ak + bk + ck]3/[a + b + c)2 → k3[a + b + c]3/[a + b + c)2
= k3(a + b + c)
L.H.S. = R.H.S. =
Hence proved.
The "X" comes from the Greek letter Chi, which is the which became Christ in English. The suffix -mas is from the Latin-derived Old English word for Mass.