Necessary and Sufficient Conditions in Mathematics

Suppose a is the condition and b is the conclusion:

If b can be deduced from a, then a is a sufficient condition for b,

B can be deduced from A, and A cannot be deduced from B, then A is a necessary and sufficient condition for B;

If a can be deduced from b, then a is a necessary condition for b,

If B cannot be deduced from A and A can be deduced from B, then A is a necessary and sufficient condition for B;

If A can't deduce B and B can't deduce A, then A is neither sufficient nor necessary for B.

If B can be deduced from A and A can be deduced from B, then A and B are necessary and sufficient conditions for each other.

Simply put:

A conclusion can be deduced from a condition, but this condition cannot be deduced from a conclusion. This condition is a sufficient (unnecessary) condition.

If conditions can be derived from conclusions, but conclusions cannot be derived from conditions. This condition is necessary (insufficient).

If we can deduce two conditions from the conclusion, we can also deduce the conclusion from the conditions. This condition is necessary and sufficient.

For example, "X=0, Y=0" is a necessary and sufficient condition for "X+Y=0".

"X+Y=0" is a necessary and sufficient condition for "X=0, Y=0".

"X+Y=0" is a necessary and sufficient condition for "X=-Y".

"X≠0, Y≠O" is an insufficient and unnecessary condition of "X+Y=0".

" sinα& gt； Sinβ "is" α > The condition of β "is neither sufficient nor necessary.

Another example is:

If two triangles are congruent, then the areas of the two triangles are equal.

So "two triangles are congruent" is a sufficient condition for "two triangles are equal in area".

But two triangles with equal areas are not necessarily the same, so

"Two triangles are congruent" is a necessary and sufficient condition for "two triangles are equal in area",

but

A, b pushes AB