Chapter 1: Q. 1 (page 97)

Determine whether each of the following statements about real numbers is true or false, and why.
Part (a): For all a, there exists some b such that $b = a^{2}$ .
Part (b): For all a, there exists some b such that $a = b^{2}$ .
Part (c): For all a, there exists some b such that $b = a + 1$ .
Part (d): For all integers a, there exists some integer b such that if $x \geq a$ , then $x > b$ .
Part (e): For all integers a, there exists some integer bsuch that if $x > a$ , then $x = b$ .

Short Answer

Expert verified

Part (a): The statement is true.

Part (b): The statement is false.

Part (c): The statement is false.

Part (d): The statement is true.

Part (e): The statement is true.

Step by step solution

Part (a) Step 1. Determine whether the statement is true or false.

Assume the case to be ${(- a)}^{2} = b a^{2} = b$ .

Thus, the statement is true.

Part (b) Step 1. Determine whether the statement is true or false.

Assume the case to be ${(- b)}^{2} = b b^{2} = b$ .

Thus, the statement is false.

Part (c) Step 1. Determine whether the statement is true or false.

Consider $f (x) = x + 1$ .

The function is one-one.

Thus, the statement is false.

Part (d) Step 1. Determine whether the statement is true or false.

Consider the given question,

'a' and 'b' both are real numbers.

If $x \leq a$ then definitely there will exist some 'b' such that $x > b$ .

Thus, the statement is true.

Part (e) Step 1. Determine whether the statement is true or false.

Consider the given question,

$a = 4$

If $x \geq a$ then $x = 6, 10, . . .$ and it is given that 'b' is a real number.

Hence, for all integers a, there exists some integer b such that $x \geq a$ then $x = b$ .

Thus, the statement is true.

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Short Answer

Step by step solution

Part (a) Step 1. Determine whether the statement is true or false.

Part (b) Step 1. Determine whether the statement is true or false.

Part (c) Step 1. Determine whether the statement is true or false.

Part (d) Step 1. Determine whether the statement is true or false.

Part (e) Step 1. Determine whether the statement is true or false.

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