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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 thatb=a2.

Part (b): For all a, there exists some b such that a=b2.

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鈮�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

01

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

Assume the case to be -a2=ba2=b.

Thus, the statement is true.

02

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

Assume the case to be -b2=bb2=b.

Thus, the statement is false.

03

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

Consider fx=x+1.

The function is one-one.

Thus, the statement is false.

04

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鈮�a then definitely there will exist some 'b' such that x>b.

Thus, the statement is true.

05

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

Consider the given question,

a=4

If x鈮�athen 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鈮�athen x=b.

Thus, the statement is true.

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