Introductory Real Analysis Trivia Quiz

Answer: N is a subset of Z, which is a subset of R.

Natural numbers are also known as counting numbers. Real numbers are considered as the universal set that includes integers, rational numbers and irrational numbers.

Answer: Infinite axiom

The name of these three axioms can be memorized by the mnemonics CAO. The algebraic axioms deal with addition, subtraction, multiplication and division of real numbers.

Meanwhile, the completeness axioms refer to maximum and minimum values of a set of numbers. On the other hand, the order axioms lay the rule for the inequalities of two or more integers.

Answer: Rational

Notice that 1 can be written as 1/1, which is a ratio of 2 integers. Therefore, 1 is a rational number. In addition, every integer {..., -2, -1, 0, 1, 2, ...} is a rational number.

Answer: Contradiction

First, we assume that surd 2 is a rational number, m/n, where m and n are positive integers with a greatest common divisor of 1, namely gcd (m, n) = 1.

Squaring both sides, we will get 2 = (m^2)/(n^2). Then, m^2 = 2n^2. Since m^2 is a multiple of 2n^2, then m^2 must be an even number. This implies that m itself is also an even number, since the square of any even number is also even.

So, after finding out m is even, we can express m = 2k. Substituting m = 2k into the equation, we will get (2k)^2 = 2n^2, that is 4k^2 = 2n^2, which can be simplified to 2k^2 = n^2. By using the same argument, we find out that n^2 and n are even also.

We find out that both m and n are even. So, gcd (m, n) is a multiple of 2. This contradicts with our previous assumption gcd (m, n) = 1. We then arrive at our conclusion that surd 2 is indeed an irrational number.

Answer: Supremum

The supremum of a set of numbers defined on an interval is also known as the least upper bound, lub.
On the contrary, the infimum of a set of numbers is also known as the greatest lower bound, glb.

Answer: Non-negative

A very famous inequality that is derived from the concept of this absolute value is the triangle inequality, which states that |a + b| is less than or equals to |a| + |b|.

Answer: 2

Notice that 1 is less than 1.23 which is less than or equals to 2. So, the correct answer is 2.

Answer: Even

We can prove this statement. Let n be the first integer. So the second integer is (n + 1). Observe that n (n + 1) = n^2 + n.

Now, if n is odd, then n^2 is odd also. Therefore, n^2 + n is an addition of 2 odd numbers which will end up in an even number.

If n is even, n^2 is also even. The addition of n^2 + n, which is the addition of 2 even numbers will also end up in an even number.

Answer: 0

When a number r is divided by positive infinity, you can imagine r is being divided by a very big number, let say 1, 000, 000, 000. Let say r = 1. So, the answer obtained will be very small, and this answer approaches 0.

Answer: True

A complex number is such as 2+3i, where i is the square root of -1. Here, the imaginary number is 3i, while 2 is the real number. Both of them make up a complex number.
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