Learn – Algebra foundations – Sets of numbers

Set representations, relations, operations

Exercises

Problem set

Represent the following sets in Venn diagrams.

  1. A = \{Alex, Rita, Parul\}, B = \{Brian, Rita, Jane\}
  2. P = \{1,2,3,4,5\}, Q = \{1,3,5,7,9\}
  3. M = \{Romeo, Juliet, Layla, Majnun, Akbar\}, N = \{Romeo, Majnun\}, P = \{Juliet, Layla\}, Q = \{Amar, Akbar, Anthony\}
  4. A = \{1,2,3,4,5,6\}, B = \{1, 3, 5\}, C = \{2,4,6\}, D = \{3, 6, 9, 12\}
  5. A = \{0,-1,1,-2,2,-3,3\}, B = \{0,1,2,3\}, C = \{0,-1,-2,-3\}, D = \{1,2,3,4\}

Problem set

Assume A = \{1,2,3,4,5\}, B = \{1,3,5\}, C = \{2,4,6\}. For each of the following, indicate if the statement is true or not.

  1. B\subseteq A
  2. C\subseteq A
  3. A\supseteq B
  4. A\supseteq C
  5. A\subseteq\mathbb{N}
  6. A\supseteq\mathbb{N}

Problem set

  1. If A = \{1,2,3\}, B = \{2,3,4\}, find A \cup B and A \cap B.
  2. If E = \{\cdots,-6,-4,-2,0,2,4,6,\cdots\}, O = \{\cdots,-5,-3,-1,1,3,5,\cdots\}, find E \cup O and E \cap O.

Natural numbers and integers

Exercises

Problem set

For each of the following, indicate if the statement is true or not.

  1. 5 \in \mathbb{N}
  2. -245 \in \mathbb{N}
  3. 10203040506070 \in \mathbb{N}
  4. 12345678 \in \mathbb{Z}
  5. -12345678 \in \mathbb{Z}

Problem set

For each of the following, indicate if the statement is true or not.

  1. \mathbb{N} \subseteq \mathbb{Z}
  2. \mathbb{Z} \subseteq \mathbb{N}
  3. \mathbb{Z} \subseteq \mathbb{Q}
  4. \mathbb{R} \subseteq \mathbb{Q}
  5. \mathbb{Q} \subseteq \mathbb{R}

Problem set

Represent the following sets in Venn diagrams.

  1. \mathbb{N}, \mathbb{Z}
  2. \mathbb{N}, \mathbb{Z}, \mathbb{Q}
  3. \mathbb{Z}, \mathbb{N}, \mathbb{R}, \mathbb{Q}

Problem set

Express the following in set builder notation.

  1. Set of all integers less than -1000.
  2. Set of all natural numbers from 10 through 1000.
  3. Set of all integers from -200 through 100.
  4. Set of all natural numbers except 33.
  5. Set of all integers except 0.

Problem set

For each of the following, answer if it is true or false.

  1. If A = \{x\,|\,x\in\mathbb{N}, x \ge 179\}, then 179 \in A.
  2. If B = \{a\,|\,a\in\mathbb{Z}, a \ge -8\}, then -10 \in B.
  3. If C = \{y\,|\,y\in\mathbb{N}, y \le 1000\}, then 3.8 \in C.
  4. If D = \{p\,|\,p\in\mathbb{Z}, p \le -1\}, then -10.1 \in D.

Problem set

Express the following in set builder notation.

  1. Set of all natural numbers less than 9000 that are divisible by 10.
  2. Set of all integers whose squares are less than 10000.
  3. Set of all odd natural numbers.
  4. Set of all perfect squares. Note that, a perfect square is the square of an integer.
  5. Set of all perfect squares that are between 100 and 1000000.

Problem set

Evaluate the following.

  1. \mathbb{N} \cup \mathbb{Z}
  2. \mathbb{N} \cap \mathbb{Z}

Rational numbers and real numbers

Exercises

Problem set

For each of the following, indicate if the statement is true or not.

  1. -9/2 \in \mathbb{Q}
  2. 5 \in \mathbb{Q}
  3. -8 \in \mathbb{Q}
  4. 0.34 \in \mathbb{Q}
  5. -17.293 \in \mathbb{Q}
  6. 0.965317317317317317317317\cdots \in \mathbb{Q}
  7. 0 \in \mathbb{Q}
  8. 0.12122122212222122222\cdots \in \mathbb{Q}
  9. \sqrt{2} \in \mathbb{Q}

Problem set

Justify that the following are rational numbers.

  1. 105.218
  2. 2.343434\cdots
  3. -56.789789789789\cdots
  4. 3.82323232323\cdots
  5. -12.60315315315315\cdots
  6. 0.99999\cdots

Problem set

For each of the following, indicate if the statement is true or not.

  1. 0.965\overline{343} \in \mathbb{R}
  2. 0.12122122212222122222\cdots \in \mathbb{R}
  3. \sqrt{2} \in \mathbb{R}
  4. \frac{0}{4} \in \mathbb{R}
  5. \frac{4}{0} \in \mathbb{R}
  6. \pi \in \mathbb{R}
  7. \sqrt{-1} \in \mathbb{R}

Problem set

Express the following in set builder notation.

  1. Set of all rational numbers between -10 and 11.
  2. Set of all real numbers between -200 and -100.
  3. Set of all real numbers that are either less than -200 or greater than 200.

Problem set

For each of the following, answer if it is true or false.

  1. If E = \{x\,|\,x\in\mathbb{Q}, x \le 5\}, then -10.37 \in E.
  2. If F = \{a\,|\,a\in\mathbb{Q}, a \ge -10\}, then 1.2\overline{345} \in F.
  3. If G = \{y\,|\,y\in\mathbb{Q}, y \le 20\}, then 0.01011011101111\cdots \in G.
  4. If H = \{p\,|\,p\in\mathbb{R}, p \le 50\}, then 1.234234423444234444\cdots \in H.

Problem set

Express the following in set builder notation.

  1. Set of all real numbers whose absolute value is less than 5.
  2. Set of all real numbers whose absolute value is less than or equal to 11.
  3. Set of all real numbers whose absolute value is greater than 7.
  4. Set of all real numbers whose absolute value is greater than or equal to 100.
  5. Set of all real numbers whose absolute value is less than or equal to 0.
  6. Set of all real numbers whose absolute value is less than 0.

Problem set

Evaluate the following.

  1. \mathbb{Q}\cap\mathbb{Z}
  2. \mathbb{Q}\cap\mathbb{I}, where \mathbb{I} represents the set of irrational numbers.
  3. \mathbb{Q}\cup\mathbb{I}, where \mathbb{I} represents the set of irrational numbers.
  4. \mathbb{R}\cup\mathbb{N}
  5. \mathbb{R}\cap\mathbb{Q}

Problem set

Evaluate the following.

  1. \{x\,|\,x\in\mathbb{R}, -7\le x\le -2\}\cap\{x\,|\,x\in\mathbb{R}, -5<x<5\}
  2. \{x\,|\,x\in\mathbb{R}, -7\le x\le -2\}\cup\{x\,|\,x\in\mathbb{R}, -5<x<5\}
  3. \{y\,|\,y\in\mathbb{Q}, -7\le y\le -2\}\cap\{x\,|\,x\in\mathbb{Q}, 2\le x\le 7\}
  4. \{y\,|\,y\in\mathbb{Q}, -7\le y\le -2\}\cup\{x\,|\,x\in\mathbb{Q}, -4< x< 0\}
  5. \{y\,|\,y\in\mathbb{R}, -7\le y\le -2\}\cap\{x\,|\,x\in\mathbb{R}, -2\le x\le 7\}
  6. \{y\,|\,y\in\mathbb{R}, -7\le y\le -2\}\cap\{x\,|\,x\in\mathbb{R}, -2< x< 7\}

Problem set

Evaluate the following.

  1. \{x\,|\,x\in\mathbb{R}, -7\le x\le -2\}\cap\{x\,|\,x\in\mathbb{Q}, -5<x<5\}
  2. \{y\,|\,y\in\mathbb{R}, -7\le y\le -2\}\cap\{x\,|\,x\in\mathbb{Z}, -4\le x\}
  3. \{y\,|\,y\in\mathbb{Q}, -7\le y\le -2\}\cap\{-5,-4,-3,-2,-1,0\}
  4. \{y\,|\,y\in\mathbb{R}, -7\le y\le -2\}\cap\{2,3,4,5,6,7\}

Problem set

Show the following on the number line.

  1. \{x\,|\,x\in\mathbb{R}, -7\le x\le -2\}\cap\{x\,|\,x\in\mathbb{R}, -5<x<5\}
  2. \{x\,|\,x\in\mathbb{R}, -7\le x\le -2\}\cup\{x\,|\,x\in\mathbb{R}, -5<x<5\}
  3. \{y\,|\,y\in\mathbb{R}, -7\le y\le -2\}\cap\{x\,|\,x\in\mathbb{R}, -4< x< 0\}
  4. \{y\,|\,y\in\mathbb{R}, -7\le y\le -2\}\cup\{x\,|\,x\in\mathbb{R}, 2\le x\le 7\}

Problem set

Express the following in interval notation.

  1. \{x\,|\,x\in\mathbb{R}, -5\le x \le -1\}
  2. \{x\,|\,x\in\mathbb{R}, -2< x < 6\}
  3. \{x\,|\,x\in\mathbb{R}, -7\le x < -2\}
  4. \{x\,|\,x\in\mathbb{R}, -11 < x \le -5\}
  5. \{x\,|\,x\in\mathbb{R}, x \ge 5\}
  6. \{x\,|\,x\in\mathbb{R}, x > -3\}
  7. \{x\,|\,x\in\mathbb{R}, x \le 4\}
  8. \{x\,|\,x\in\mathbb{R}, x < -1\}

Problem set

Express the following in interval notation.

  1. \{y\,|\,y\in\mathbb{R}, -7\le y < -2\}\cap\{y\,|\,y\in\mathbb{R}, -4< y < 2\}
  2. \{x\,|\,x\in\mathbb{R}, -7< x< -2\}\cup\{x\,|\,x\in\mathbb{R}, -4< x < 2\}
  3. \{x\,|\,x\in\mathbb{R}, -7< x< -2\}\cup\{x\,|\,x\in\mathbb{R}, -1< x \le 2\}
  4. \{x\,|\,x\in\mathbb{R}, -7< x< -2\}\cup\{x\,|\,x\in\mathbb{R}, x > -1\}
  5. \{x\,|\,x\in\mathbb{R}, -7< x< -2\}\cup\{x\,|\,x\in\mathbb{R}, x > -3\}
  6. \{x\,|\,x\in\mathbb{R}, x\ge 0\}\cup\{x\,|\,x\in\mathbb{R}, x< 0\}
  7. \mathbb{R}\cap\{x\,|\,x\in\mathbb{R}, x< 0\}

Miscellaneous

Exercises

Problem set

Assume A is some set and U is the universal set. For each of the following, indicate if the statement is true or false.

  1. A\subseteq U
  2. \phi\subseteq A
  3. A\subseteq A
  4. U\supseteq A
  5. \phi\supseteq A

Problem set

  1. Is every number in the collection of natural numbers present in the collection of whole numbers?
  2. Give five numbers that are present in the collection of integers, but not in the collection of whole numbers.
  3. Give two numbers that are commonly present in the collection of integers and the collection of natural numbers.
  4. Give a number that is present in the collection of integers and the collection of whole numbers, but not in the collection of natural numbers.
  5. Is there a number that is present in the collection of whole numbers, but not in the collection of integers?
  6. Draw a diagram showing how the collections of natural numbers, whole numbers and integers contain one another.