Yolisa – Page 2 – Mathemafrica

0.2. Subsets

https://giphy.com/gifs/infinite-boxes-vG1Dgq3JRXLMc

Consider a set $A = \{2, 3, 7\} \text{ and } B = \{2, 3, 4, 5, 6, 7\}.$ Note that every element in set A is also found in set B, however, the reverse is not true (B contains elements 4, 5 and 6 which are not in A)

Consider another case, $A = \{2n: n \in \mathbb{ N }\} = \{ 0, 2, 4, 6, ... \} \text { and } B = \mathbb{Z} = \{ ..., -2, -1, 0, 1, 2, ... \}.$ Again, we can see that every element in set A is also found in set B and similarly, everything in B cannot be found in set A. B contains negative and odd integers, which are not in A.

To describe this phenomena, mathematicians defined subsets:

$def^n$ Suppose A and B are sets. If every element in A is an element of B, then A is a subset of B and we denote this as $A \subseteq B$

If B is not a subset of A, as in the above cases, then there exists at least one element, say $x \in B \text{ such that } x \notin A. \text{ We denote this as } B \subsetneq A$

e.g.1. $\{2, 3, 5, 7, ... \} \subseteq \mathbb{ N }$ but $\{\frac{1}{3}, 2, 5, 7, ... \} \subsetneq \mathbb{ N }$ since $\frac{1}{3} \in \mathbb{ Q }$

e.g.2. $\mathbb { N } \subseteq \mathbb { Z } \subseteq \mathbb{ Q } \subseteq \mathbb{ R }$

e.g.3. $(\mathbb{ R } \times \mathbb{ N }) \subseteq (\mathbb{ R } \times \mathbb{ R })$ since $(\mathbb{ R } \times \mathbb{ N }) = \{(x, y): x \in \mathbb{ R }, y \in \mathbb{ N }\}$ and $( \mathbb{ R } \times \mathbb{ R }) = \{ (x, y): x \in \mathbb{ R }, y \in \mathbb{ R } \}$ Hint: look at what sets y is in

Every set is a subset of itself :

e.g.1.

…

0.1 Sets

If like me, you’ve spent most of your mathematical high school years introduced to basic sets at the beginning of the year from Grades 8 to 12, then I think you’d agree that sets was one of the quickest and easiest sections we traditionally did. We would quickly recap the same fundamental properties of sets before moving onto more interesting topics, usually algebra. The section would go a little bit like this:

define the differences between whole and natural numbers, integers, rational numbers and real numbers
define the differences between unions, intersections and complements, usually through the understanding of Venn-diagrams
use set builder notation (introducing algebra through this)

If like myself, you truly believed that this was as complicated as sets could ever get, then you, dear reader, like my former-myself, are in for a treat. In university, we build on these basic ideas and have a more in depth understanding about the importance of sets and their greater role in the scheme of mathematics.…

About Yolisa

0.2. Subsets

0.1 Sets