Problem 2. (a) Let P(z) be a statement whose truth value depends on z. An ezampl

Question

Problem 2. (a) Let P(z) be a statement whose truth value depends on z. An ezample is a value of z that makes P(x) true, and a counterezample is a value of x that makes P(x) false. Fill in the blank spaces with "is true", "is false'", or "nothing" as appropriate: "3z s.t. P(x)" An example proves A counterexample proves In (b) - (h), determine whether the given statement is true or false, and briefly justify your answer. (b) Every integer is even or odd (c) Every integer is even or every integer is odd. (d) Some rational numbers are real numbers (e) Every complex number is real or imaginary. (f) There is x € R such that for every R, y z. (g) For every y R there is x R such that y x.

Explanation / Answer

(a)

Having P(x) true for one x, just ensures that P(x) is true for atleast one x but not for x.

Having P(x) false for one x, just ensures that P(x) for all x statement is false.

(b) TRUE. All the integers which on dividing by 2 gives 0 are even and rest all of them are odd. so every integers has to be in one of these two sets.

(c) FALSE. The statement says that either all the integers are even or all the integers are odd. Which is not true. Because we do have examples of both even and odd numbers like 2 is even and 3 is odd.

(d) TRUE. All rational numbers are real numbers so statement that some rational numbers are real is true.

(e) TRUE. A complex number can be both real and imaginary depending upon its real and imaginary part.

(f) FALSE. e.g. x = 2, so it is not necessarily y < 2

(g) TRUE. for every value y (let for an example y = 2) , there are real values of x such that x > 2

for all x, P(x) there exists x, s.t. P(x) An example proves nothing is true a counterexample proves is false nothing

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