11. Predicates P and Q are defined below. The domain of discourse is the set of

Question

11. Predicates P and Q are defined below. The domain of discourse is the set of all positive integers. P(x): x is prime Q(x): x is a perfect square Are the following logical expressions propositions? If the answer is yes, indicate whether the statement is true or false. a. P(x) b. P(2) 12. Consider the following statements in English. Write a logical expression with the same meaning. The domain of discourse must be the set of all real numbers. There is a number whose cube is equal to -2. a. b. The reciprocal of every number between O and 1 is greater than 1. 5

Explanation / Answer

11. Proposition means a statement which means either true or false.

   a) P(x) we can't say true or false for this so not a proposition
   b) P(2) it is true . So it is a proposition
   c) x(Q(x)-> ¬P(x)) - It is true . It is a proposition

12.

a. Considering a Predicate as follows:

   Cube(x,y) - y is the cube root of x

       x(Cube(x,-2))    x belongs to all real numbers

b. We can have following predicates:

    R(x,y) - y is the reciprocal of x
    L1(x) - x is less than 1
    G1(x) - x is more than 1
    L0(x) - x is less than 0
    G0(x) - x is more than 0

    So now we can write

     x y(G0(x)L1(x)R(x,y)-->G1(y))

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