Symbol for the number of elements in a set

Question

I would like to express below concisely and mathmatically

1. TP is a set of real numbers

2. $$C = \{ x \in TP : x > threshold \}$$

3. c_count = len(C)

Basically, in English, I want to count the number of numbers in TP is greater than threshold

Answer 1

Since $C$ is defined as a subset of $TP$ based on the premise of being greater than some threshold, then, if $C$ is finite, you can refer to "cardinality" - it is a measure of how many elements are in a set. It gets murkier for infinite sets, but for finite sets, the cardinality of a set is just the number of elements in the set.

How would one denote cardinality? There are several conventions I've seen:

• $|C|$ (using absolute value signs)
• $\#C$ (using a number sign)
• $\text{card}(C)$ (as a function itself, effectively)

Answer 2

With $C$ defined as above, $|C|$, the cardinality of $C$, would represent the number of elements $x$ in $TP$ such that $x>\text{threshold}$. Written out fully: $$|C|=|\{x\in TP\mid x>\text{threshold}\}|$$