set notation confusion - object t is a member of closed interval set?

Question

I'm reading through an introduction to basic set theory and have encountered the following ..

$t\in [a, b] \Leftrightarrow a \leq t \:\&\: t \leq b$

I've been trying to figure the above in combination with the Wikipedia list of mathematical symbols, but I'm still not clear how to read this. I understand the part to the right of the "if and only if ..." double headed arrow, but the bit to the left I'm not sure about.

Simple explanations will be preferred over complex ones.

Answer 1

You read it as: “$t$ belongs to the closed interval $[a,b]$ if and only if $a$ is smaller than or equal to $t$ and $t$ is smaller than or equal to $b$”.

Answer 2

$t$ is the variable $t$. It represents supposedly a real number.

$\in$ means "is a member of" a set. The next thing to follow will be a set of real numbers and the statement is that the number represented by $t$ is an element of the set.

$[a,b]$ is a set of all the real numbers that are between $a$ and $b$ inclusively. Or in other words the set of all real numbers that are both $\ge a$ and $\le b$.

So $t \in [a,b]$ means $t$ is the variable representation of a real number that is in the set of all real numbers that are greater or equal to $a$ or less than or equal to $b$.

Note: $t \in [a,b]$ is another way of stating $a \le t \le b$.

That's the LHS.

$\iff$ means "if and only if". the LHS is true precisely and only when the RHS is true. If $LHS \iff RHS$ then we say "LHS" and "RHS" are equivalent as they are, as far as the rules of the universe allows us to accept reality, the exact same thing. You simply can not have one without the other. We often use these for making definitions. A konkle is defined to be a green blickle. So Jim is a konkle if he is a green blickle and if Jim is a green blickle he is a konkle.

$a$ is a constant real number $a$

$\le$ is less than or equal to.

$t$ is a variable for some real number

& is "and"

$t$ is our variable again and and $\le$ is less than or equal to again.

$b$ is another constant.

So RHS is $a$ is less than or equal to $t$ and $t$ is less than or equal too $b$.

So the statement:

$t \in [a,b] \iff a \le t \& t\le b$ means:

The real number $t$ is a real number in the set of all numbers between $a$ and $b$ inclusively if an only if $a$ is less than or equal to $t$ and $t$ is less than or equal to $b$.

.... which is .... obvious.

Basically this is simply defining what the notation $t \in [a,b]$ means. $t$ is a member of the set $[a,b]$ if and only if $a \le t \le b$.

Another way of putting it is $[a,b]$ is defined to be the set of all real numbers so that $a \le $ the real number $\le b$.