Set Theory and Rational Numbers Quiz

<p>At least one of m or n must be zero. (C)</p> Signup and view all the answers

<p>The operation is addition as described by p r ps + qr. (A)</p> Signup and view all the answers

<p>I does not cover all integers greater than or equal to m. (A)</p> Signup and view all the answers

<p>The set of rational numbers excluding both 0 and 1. (B)</p> Signup and view all the answers

<p>If m.n = p.n, then m and p must be equal only if n ≠ 0. (C)</p> Signup and view all the answers

<p>Finite sets can be counted and listed without repetition. (B)</p> Signup and view all the answers

<p>q will be greater than p. (C)</p> Signup and view all the answers

<p>n.m = m.n (C)</p> Signup and view all the answers

<p>It defines multiplication on Q. (D)</p> Signup and view all the answers

<p>a is a lower bound but not the infimum if β &gt; a. (A)</p> Signup and view all the answers

<p>It defines a method to find a lower bound. (A)</p> Signup and view all the answers

<p>q is a rational number less than 2. (D)</p> Signup and view all the answers

<p>As pr/qs where p/q and r/s are rational numbers. (D)</p> Signup and view all the answers

<p>Every open interval contains a member of D. (A)</p> Signup and view all the answers

<p>The Density Theorem. (C)</p> Signup and view all the answers

<p>It may be bounded above but lacks a rational supremum. (B)</p> Signup and view all the answers

<p>There exists a unique real number β that is the supremum of S. (D)</p> Signup and view all the answers

<p>qa &lt; p &lt; qa + q(b − a). (A)</p> Signup and view all the answers

<p>Every positive integer n can be found such that nε &gt; ρ. (B)</p> Signup and view all the answers

<p>p &lt; √2. (D)</p> Signup and view all the answers

<p>There exists a bound but no least upper bound that is rational. (C)</p> Signup and view all the answers

<p>For every $orall eta &gt; 0$, there exists an integer K such that $|s_n - s_m| &lt; eta$ if $m, n oldsymbol{≥} K$. (B)</p> Signup and view all the answers

<p>Lim sup and lim inf must be equal for the limit to exist at a finite value. (B)</p> Signup and view all the answers

<p>The sequence is bounded. (D)</p> Signup and view all the answers

<p>If there exists an integer N such that $|s_n - s_m| &lt; ε$ for m, n ≥ N. (B)</p> Signup and view all the answers

<p>s and s must be finite and equal for the limit to exist. (D)</p> Signup and view all the answers

<p>The sequence must be strictly increasing. (C)</p> Signup and view all the answers

<p>If both limits exist and are equal, the sequence converges. (B)</p> Signup and view all the answers

<p>The terms of the sequence are limited to a certain range. (A)</p> Signup and view all the answers

<p>To illustrate that both inequalities $y_n &lt; x$ and $y_n &gt; x$ lead to a contradiction (A)</p> Signup and view all the answers

<p>It applies universally regardless of the signs of $a$ and $b$. (C)</p> Signup and view all the answers

<p>$a ≥ 0$ and $b ≤ 0$ (D)</p> Signup and view all the answers

<p>It establishes a lower bound for the absolute difference of two numbers. (D)</p> Signup and view all the answers

<p>It creates an upper bound that contradicts $y$ being the least upper bound. (C)</p> Signup and view all the answers

<p>Both $a$ and $b$ are positive. (A), Both $a$ and $b$ are negative. (D)</p> Signup and view all the answers

<p>It establishes a relationship between power functions of different bases. (B)</p> Signup and view all the answers

<p>It shows that $y - k$ cannot exceed $y$, contradicting the definition of upper bounds. (B)</p> Signup and view all the answers

<p>It is the least upper bound that satisfies certain conditions. (A)</p> Signup and view all the answers

<p>It must have a supremum that is a real number. (D)</p> Signup and view all the answers

<p>It is less than or equal to every element in the set. (A)</p> Signup and view all the answers

<p>There exists an element in the set that exceeds it. (A)</p> Signup and view all the answers

<p>If one supremum exists, no other can satisfy both properties. (B)</p> Signup and view all the answers

<p>There exists an integer n such that nε &gt; ρ. (B)</p> Signup and view all the answers

<p>ρ is an upper bound of the set S where S = {x = nε, n ∈ Z}. (C)</p> Signup and view all the answers

<p>It is the supremum of S. (D)</p> Signup and view all the answers

A non-empty subset of Natural Numbers (N) such that if 'n' is in the set and 'm' is less than or equal to 'n', then 'm' is also in the set.

A set that is either empty or has a one-to-one and onto mapping (bijective) to an initial segment of natural numbers.

A set that is not finite. It cannot be put into a one-to-one correspondence with any initial segment of natural numbers.

For any natural numbers 'm' and 'n', 'm times n' is equal to 'n times m'.

For any natural numbers 'm', 'n', and 'p', (m times n) times p is equal to m times (n times p).

If 'm times n' is equal to 'p times n', and 'n' is not zero, then 'm' equals 'p'.

The product of any number and zero is zero.

Multiplying a number by one does not change the number.

The set of numbers that can be expressed as a fraction m/n, where m and n are integers, and n is not zero.

Represents a set of rational numbers.

The set A contains rational numbers p where p² is less than 2.

A value 'a' which is less than or equal to every element in a set.

The greatest lower bound of a set (in this case B).

The process to identify a rational number 'q' such that p < q and 'q' represents a rational number where q² will be less than 2 if p² is less than 2.

A set with two operations (usually addition and multiplication) that follow specific properties.

The set of rational numbers excluding zero and one.

A nonempty set of real numbers that is bounded above has a supremum (least upper bound).

The real number system is complete because every nonempty set of real numbers that is bounded above has a least upper bound (supremum).

The smallest upper bound of a set. It's the smallest number that is greater than or equal to all elements in the set.

For any positive numbers ρ and ε, there exists an integer n such that nε > ρ.

A set of real numbers is 'bounded above' if there exists a real number that is greater than or equal to all elements in the set.

A proof method where you assume the opposite of what you want to prove and then show that this assumption leads to a logical contradiction.

Does the set of rational numbers (Q) form a subset of the set of real numbers (R)?

Is there only one supremum for a nonempty set of real numbers that is bounded above?

A set is dense in another set if every open interval of the latter set contains a member of the former set.

Given any two real numbers, there always exists a rational number between them.

A set is bounded above if there exists a real number (an upper bound) that is greater than or equal to all elements in the set.

The supremum of a set is the smallest real number that is greater than or equal to all elements in the set.

A set is complete if every bounded above subset has a least upper bound (supremum) within the set.

There are sets of rational numbers that are bounded above but do not have a least upper bound (supremum) within the set of rational numbers.

The square root of 2 cannot be expressed as a ratio of two integers.

The smallest number that is greater than or equal to all elements in a set.

A method of proof where you assume an inequality (e.g., 'a < b') and show that it leads to a contradiction.

An equation that holds true for all values of the variables involved.

The absolute value of the sum of two numbers is less than or equal to the sum of their absolute values.

The distance of a number from zero, always positive.

The absolute value of the difference of two numbers is greater than or equal to the difference of their absolute values.

A theorem that is a direct consequence of another theorem.

A sequence {sn} converges to a limit 's' if, for any small positive number (ε), we can find a point in the sequence (sn) such that all subsequent terms are within ε distance from 's'.

The largest limit point of a sequence {sn}. It represents the 'highest' value that the terms of the sequence can get arbitrarily close to.

The smallest limit point of a sequence {sn}. It represents the 'lowest' value that the terms of the sequence can get arbitrarily close to.

A sequence where, for any small positive number (ε), we can find a point in the sequence (sn) such that all terms after that point are within ε distance from each other.

A sequence {sn} converges if and only if it is a Cauchy sequence.

The sum of the terms of a sequence.

A sequence for which there exists a number that is larger than or equal to every term in the sequence.

A sequence {sn} converges to a limit 's' if the limit superior and limit inferior of the sequence both equal 's'.