26 Let POr y be the proposition y where the variables range

26. Let POr, y) be the proposition y, where the variables range over the real numbers. Determine the truth value of each of the following propositions. (These particular propositions will be important in Section 3.3.) (a) var: Pac, z) (c) var: Vy: (P(a, y) POy, ar))

Solution

Solved the first four sub-parts, please post one more question to get the remaining answers

(a) The first statement is TRUE

since for every x, x<=x satisfies the equation

Hence the statement isTRUE

(b) The second statement is FALSE

since for every x, x<=x satisfies the equation and the statement is the inverse of true statement

Hence the statement is FALSE

(c) The third statement is FALSE

since P(x,y) doesn\'t imply P(y,x)

x<=y => y<=x, since if x is less than or equal to y, then y can\'t be less than or equal to x

Hence the statement is FALSE

(d) The statement is TRUE

since the first statement implies x<=y and second statement implies y<=x, the two things can hold if and only if (x=y)

Hence the given statement is TRUE