Write the following statements in predicate form using logic

Write the following statements in predicate form, using logical operators , , ¬, and quantifiers , . Below Z+ denotes all positive integers {1, 2, 3, . . .}.

(a) The square of a positive integer is always bigger than the integer.

(b) There is no integer solution to the equation x = x + 1.

(c) The absolute value of an integer is not necessarily positive.

(d) The absolute value of the sum of two integers does not exceed the sum of the absolute values of those integers.

Solution

a) means for all x.

x(x² > x), x denotes Z+.

b) means there exists, ¬ means negation. So, ¬ means there does not exist.

¬x (x = x + 1), x denotes Z+.

c)x (|x| <= 0), where x is an integer.

d)x,y (|x+y| <= |x|+|y| )

Here < = means it is either less than or equal to.