Suppose $f$ is a real valued identically constant function $f(x)=\beta ,$ then $\left\{ x:f(x)>\alpha \right\} =% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion$. And if $\beta >\alpha$, then__________.

• A) $f$ is $1-1$
• B) $f~$is not measurable
• C) $f$ is bijective
• D) $f$ is measurable

Let $M$ be a measurable space (a sigma algebra of measurable sets) then a function $f:E\longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion$ on $\varnothing \neq E\in M$ given as $\left\{ x\in E:f(x)>a\right\}$ is measurable if $\forall a,$----------under $f$ is measurable set.

• A) inverse image of $\left( a,\infty \right)$
• B) image of $\left( x,\infty \right)$
• C) inverse image of $\left( a,\infty \right)$
• D) image of $\left( a,\infty \right)$

$${\text{Which of the following best describes a measurable function?}}$$

• A) $${\text{A function that maps measurable sets to measurable sets}}{\text{.}}$$
• B) $${\text{A function that can be represented as the union of countably many open sets}}{\text{.}}$$
• C) $${\text{A function that maps Borel sets to Lebesgue measurable sets}}{\text{.}}$$
• D) $${\text{A function that preserves the order of elements in its domain}}{\text{.}}$$

$${\text{Which of the following sets is countable and also a Borel set with measure zero?}}$$

• A) $${\text{The interval [0,1]}}$$
• B) $${\text{The set of irrational numbers}}$$
• C) $${\text{The set of real numbers}}$$
• D) $${\text{The set of natural numbers}}$$

What is a characteristic function in mathematics?

• A) A function that describes the characteristics of a set
• B) A function that calculates the average of a set
• C) A function that assigns values to elements of a set
• D) A function that indicates membership of an element in a set

What does the characteristic function XA∩B​(x) represent?

• A) The maximum value of the characteristic functions of sets A and B.
• B) The minimum value of the characteristic functions of sets A and B.
• C) The difference of the characteristic functions of sets A and B.
• D) The sum of the characteristic functions of sets A and B.

$${\text{Let }}f:\mathbb{R} \to \mathbb{R}{\text{ defined by }}f(x) = 2.{\text{ The set }}\{ x \in \mathbb{R}:f(x) > 7\} {\text{ is equal to which of the following sets?}}$$

• A) $$\left( { - \infty ,7} \right)$$
• B) $$\left( {7,\infty } \right)$$
• C) $$\mathbb{R}$$
• D) $$\emptyset$$

What value does the characteristic function relative to a set non-emty set A take if an element belongs to the set A?

• A) Depends on the element.
• B) 1
• C) 0
• D) -1

How does monotonicity in characteristic functions help in comparing sets?

• A) It facilitates understanding of the relationship between sets.
• B) It provides insights into the functions' behavior.
• C) It helps in finding the derivative of the functions.
• D) It aids in determining continuity of the functions.

$${\text{Let }}f:\mathbb{R} \to \mathbb{R}{\text{ defined by }}f(x) = 9.{\text{ The set }}\{ x \in \mathbb{R}:f(x) > 7\} {\text{ is equal to which of the following sets?}}$$

• A) $$\left( { - \infty ,7} \right)$$
• B) $$\emptyset$$
• C) $$\mathbb{R}$$
• D) $$\left( {7,\infty } \right)$$

Which of the following statements accurately describes countable sets in relation to Borel sets?

• A) Countable sets cannot be Borel sets.
• B) Countable sets are Borel sets but may or may not have measure zero.
• C) Countable sets are always Borel sets with measure zero.
• D) Countable sets are not Borel sets with measure zero.

Let $M$ be a measurable space (a sigma algebra of measurable sets) then a function $f:E\longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion$ on $\varnothing \neq E\in M$ is measurable if the set $\left\{ x\in E:f(x)>a\right\}$ is measurable for all __________.

• A) $x\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion$
• B) $a\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion$
• C) $a\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \cup \left\{ \pm \infty \right\}$
• D) $x\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion \cup \left\{ \pm \infty \right\}$

Suppose $f$ is a real valued identically constant function $f(x)=\beta ,$ then $\left\{ x:f(x)>\alpha \right\} =$_________, if $\beta >\alpha$.

• A) $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion$
• B) $\varnothing$
• C) $\left\{ \alpha \right\}$
• D) $\left\{ \beta \right\}$

Let $x,y\in \left[ 0,1\right) ,$then the sum modulo $1;x\widehat{+}y$ of $\ x$ and $\ y$ is defined by_________.

• A) $x+y-1$
• B) $x+y$ if $x+y<1$ and $x+y-1$ if $x+y\geq 1$
• C) $x+y$ if $x+y\leq 1$ and $x+y-1$ if $x+y>1$
• D) $x+y$ if $x+y\geq 1$ and $x+y-1$ if $x+y<1$

If one set is a subset of another, what is the relationship between their characteristic functions?

• A) The larger set's characteristic function is less than the smaller set's.
• B) They are equal.
• C) They are unrelated.
• D) The smaller set's characteristic function is less than or equal to the larger set's.

What does the characteristic function of the empty set indicate?

• A) That no element belongs to the set.
• B) That the set contains all possible elements.
• C) That every element belongs to the set.
• D) That the set has infinite elements.

What property describes the characteristic function of the empty set?

• A) Exponential function
• B) Linear function
• C) Constant function
• D) Quadratic function

What does it mean if the characteristic function of set A is less than the characteristic function of set B for all elements?

• A) A is a proper subset of B
• B) A and B are equal sets
• C) A and B are disjoint sets
• D) A and B have the same cardinality

In $\left[ 0,1\right) ,$ the sum modulo $1,x\widehat{+}y$ of $\frac{3}{2}$ and $\frac{1}{2}=$___________.

• A) $3$
• B) $1$
• C) is not defined at all
• D) $\frac{1}{2}$

How many values can a characteristic function take?

• A) 1
• B) Infinite
• C) 3
• D) 2