List all the elements of the following sets:
\left\{ \text{positive multiples of } 4 \text{ that are less than } 46 \right\}
\left\{ \text{positive even integers that are less than } 16 \right\}
\left\{ \text{positive factors of } 42 \right\}
\left\{ \text{square numbers less than } 109 \right\}
\left\{ \text{numbers that can be rolled on a standard die} \right\}
\left\{ \left. x \right\vert x \text{ is a natural number less than } 5 \right\}
\left\{ \left. x \right\vert x \text{ is an odd whole number less than } 13 \right\}
\left\{ \left. x \right\vert x \text{ is an integer between } - 8 \text{ and } - 3 \text{ (not inclusive)} \right\}
Determine whether the following statements are true or false:
3 \in \left\{ \left. x \right\vert x \text{ is a rational number} \right\}
3 \in \left\{2, 4, 5, 8\right\}
1 \in \left\{9, 8, 6, 5, 1\right\}
-0.25 \in \left\{\left. x \right\vert x \text{ is a natural number}\right\}
Consider the set A = \left\{2, 4, 6, 8\right\}. Construct a set builder notation for A.
Determine whether the following statements are true or false:
-6 \notin \Z
Determine whether the following statements are true or false:
For each of the following numbers sets, state whether they approach positive infinity as the numbers increase. If not, state the set's limit.
The real numbers, \Reals.
The rational numbers, \mathbf{Q}.
The integers, \Z.
The positive integers, \Z^+.
The natural numbers, \N.
For each of the following numbers sets, state whether they approach negative infinity as the numbers decrease. If not, state the set's limit.
The real numbers, \Reals.
The rational numbers, \mathbf{Q}.
The integers, \Z.
The positive integers, \Z^+.
The natural numbers, \N.
For each of the following numbers sets, state whether they are dense.
The real numbers, \Reals.
The rational numbers, \mathbf{Q}.
The integers, \Z.
The positive integers, \Z^+.
The natural numbers, \N.
Complete the following table to indicate whether each number belongs in the number set:
| \N | \Z | \Z^+ | \mathbf{Q} | \Reals | |
|---|---|---|---|---|---|
| 5 | Y | Y | Y | Y | Y |
| -3 | N | Y | N | N | Y |
| 123 | |||||
| -94.5 | |||||
| \dfrac{3}{4} | |||||
| \sqrt{5} | |||||
| \sqrt{-1} |
Write down the next 3 trianglular numbers in this sequence:
1, 3, 6, 10, 15, ⬚, ⬚, ⬚Beginning with the eighth triangular number, write down the next 3 triangular numbers in this sequence:
36, 45, 55, 66, 78, ⬚, ⬚, ⬚
Let A = \left\{ x | \, x \text{ is a triangular number} \lt 60 \right\} and let B = \left\{ x | \, x \text{ is a positive multiple of } 3 \lt 60 \right\}.
Write all the terms in set A.
Write all the terms in set B.
Find A \cap B.
Is A \subseteq B? Explain your answer.
Is A \subseteq \N? Explain your answer.