Q. 1 - Calculus

Question

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 $x\ge a$, then $x>b$.

Part (e): For all integers a, there exists some integer b such that if $x>a$, then $x=b$.

Step-by-Step Solution

Answer

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. 

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

Assume the case to be $$\begin{gathered} {\left(-a\right)}^2=b \\ {a}^2=b \end{gathered}$$.

Thus, the statement is true.

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

Assume the case to be $$\begin{gathered} {\left(-b\right)}^2=b \\ {b}^2=b \end{gathered}$$.

Thus, the statement is false.

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

Consider $f\left(x\right)=x+1$.

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 $x\le a$ 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 $x\ge a$ 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 $x\ge a$ then $x=b$.

Thus, the statement is true.