[Arithmetic 6] The number of elements in a set. Subset

I. THE NUMBER OF ELEMENTS IN A SET

$A = \left\{ 2 \right\}$ has 1 element.

$B = \left\{ {a,b} \right\}$ has 2 elements.

$C = \left\{ {1;2;3; \ldots ;100} \right\}$ has 100 elements.

$\mathbb{N} = \left\{ {0;1;2;3; \ldots } \right\}$ has infinite elements.

* Note: The set which has no elements is called the emty set and is denoted by $\large\varnothing$.

   Example: $M=\left\{ x\in\mathbb{N} \big| x + 5 = 2 \right\} = \large\varnothing$.

* Comment: A set may have one element, many elements, infinite elements, and may be no elements.

II. SUBSET

If every element in set $E$ belongs to set $F$, then set $E$ is called the subset of set $F$, and is denoted by $E$ $\subset$ $F$ (hay $F$ $\supset$ $E$).

Example 1: Given $E = \left\{ {a,b} \right\}$, $F = \left\{ {a,b,c,d} \right\}$.

            We have $E$ $\subset$ $F$ (or $F$ $\supset$ $E$).

Example 2: 

        $M = \left\{ {1;5} \right\}$,

        $A = \left\{ {1;3;5} \right\}$,

        $B = \left\{ {5;1;3} \right\}$.

We have $M$ $\subset$ $A$, $M$ $\subset$ $B$

              $A$ $\subset$ $B$, $B$ $\subset$ $A$.

* Note: $A$ $\subset$ $B$, $B$ $\subset$ $A$ hence $A = B$.