[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: $M=\left\{ x\in\mathbb{N} / x + 5 = 2 \right\} = \large\varnothing$

* 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