What's the different between necessary, sufficient, necessary and sufficient condition?

Question

1)The range of values of "$a$", such that

$|x-2|< a$ is a necessary condition for $x^2-3x-10<0$

2)The range of values of "$a$", such that

$|x-2| < a$ is a sufficient condition for $x^2-3x-10<0$

I have found that a necessary and sufficient condition for $x^2-3x-10<0$ is $-2< x <5$
, how can I answer that two problems? and what is the different between necessary, sufficient, necessarry and sufficient condition? is there about logical mathematics ?

Answer 1

Saying that $A$ is necessary for $B$ means that $B$ implies $A$-that any time $B$ is true, so is $A$. If $B$ is false, $A$ may still be true.

Saying that $A$ is sufficient for $B$ means that $A$ implies $B$.

Saying that $A$ is necessary and sufficient for $B$ means both of the above are true, so $A$ and $B$ are either both true or both false.

Answer 2

Necessary, but not sufficient condition: $$a>0,$$ because the absolute value inequality must hava a silution.

Necessary and sufficient condition: $$x^2-3x-10<0 \Rightarrow -2<x<5 \Rightarrow -4<x-2<3 \Rightarrow |x-2|<3 \Rightarrow a\le 3;\\ 0<a\le 3.$$

Answer 3

Necessary is what it absolutely must have.

To be a dessert it is necessary that it be food.

To show if $|x-2| < a$ is necessary condition for $x^2 - 3x -10 <$ you must prove

$x^2 - 3x -10 > 0 \implies |x-2| < a$.

Sufficient is something it doesn't need to be, but if it is that's enough to show it is true.

To be a dessert it is sufficient to be ice cream.

(Not all desserts are ice cream, but all ice creams are desserts. So if you can prove something is an ice cream that is enough, or in other words, that is sufficient to prove it is a dessert.

To show if $|x-2| < a$ is necessary condition for $x^2 - 3x -10 <$ you must prove

$|x-2| < a \implies x^2 - 3x -10 > 0$.

Nescessary and sufficient is both. It be needs to be and if it is that will prove it.

To be a dessert is is necessary and sufficient to be a sweet food eaten after dinner.