Let X be a non-empty set. A one-argument propositional function over space X  (function range) is an expression α(x) in which x is an argument and which becomes true or false if we substitute any arbitrary object of the set X for the variable x.

When X is a Cartesian product of sets X₁ × ... × X_n the variable x takes as value some element of this product. We say in this case that we have an n-argument propositional function. In order to simplify our notation, we write down α(x) where x  can be both one simple variable or a vector of variables (see subsection 3.5).

Example 7.1.1

1. α(x) = (2x+x²) >0 for x ∈ Z is a propositional function dependent on one variable x over set of integer numbers. If x is replaced with an arbitrary value e.g. 5, then the function's value is true (because (10+25) is a positive number). On the other hand, when  x = -1 then α(x) value is equal to false (because (2*(-1) + (-1) ² ) < 0).
2. β(x,y) = (x² + y² ) >0 for (x,y) ∈R² is a propositional function dependent on two variables x and y over the set of real numbers. If x and y are replaced with freely chosen numbers, then we may obtain for example the following sentences "the sum of one squared and two squared is a positive number" or "zero squared plus zero squared is greater than zero", which are respectively true and false.
   

Question 7.1.1

1. Express (as shortly as you can) in words the following propositional function over real numbers : ( a * b) > (a + b).
2. Write down, using proper symbols, the following sentence "Square of the sum of two real numbers a, b is not less than the sum of its squares".

Remark The one-argument propositional function f(x) is simply a function defined over some arbitrary set X with values in the two-element Boolean algebra, f : X → {0,1} where for any x ∈ X, f(x) is a proposition in the sense of propositional calculus and takes value 1 if and only if this proposition is valid.

Propositional functions, in other words predicates, may be connected with logical functors which were introduced in the previous lecture. This process leads to new complex propositional functions which we call formulas of predicate calculus. If α(x) and β(x) are predicates then

are complex predicates or formulas.

Example 7.1.1

The following expressions are examples of complex predicates:

1. ((x-3 = 0) ∨(x+y < 4)) → (y< 1),
2. (A ∪ B ⊆ C → (A ⊆ C ∧ B ⊆ C)),
3. if Adam is the son of Andrew and Andrew likes classic music then Adam likes classic music.

Question 7.1.2 Let p(x) stand for "x is an even number",  mod₄(x) be the function which gives the remainder of division x by 4 and let " /2 " be the operation of division by 2. Write down sentence "If x is an even number and x is divisible by 4, then x divided by 2 is an even number too".

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