Please show step by step how you would do these I do not und

Please show step by step how you would do these I do not understand

1) Determine whether x(P(x)--> Q(x)) and xP(x)-->xQ(x) are logically equivalent

2) Determine whether x(P(x) <--> Q(x)) and xP(x) <-->xQ(x) are logically equivalent

Solution

The given two statements are NOT logically equivalent

In order to disprove the above statement, let us assume two statements P and Q and check the truth tables match for each function P and Q

P(x) be a function with number x greater than 100

Q(x) be a function with number x less than 0

x(P(x)--> Q(x)) --> The function is FALSE, since for every x P(x) doesn\'t implies Q(x)

xP(x)--> xQ(x) is also FALSE

Let us assume a number which is 25, then the statement will be TRUE for the second case, since both the statement are false, false-> false will be true

Whereas second statement is False

Hence they are not logically equivalent