Write the reciprocal and the anti-reciprocal of:
That a quadrilateral has 4 equal sides, is a necessary condition for that it is a square.
That a triangle has 2 equal sides, is a necessary condition for that it is isosceles.
Only if f is integrable in an I interval, f is continuous in I.
If an f function is derivable in a point then f is continuous in that point.
F is not an even function, to less that f(x)=f(-x).
If a<b then a^2< b^2.
To find the reciprocal of a statement, we need to consider the inverse of the original statement. In mathematical logic, the inverse is formed by negating both the hypothesis (the part that comes after "if") and the conclusion (the part that comes after "then").
1. Original statement: A quadrilateral has 4 equal sides -> The quadrilateral is a square.
Reciprocal: If a quadrilateral is not a square, then it does not have 4 equal sides.
2. Original statement: A triangle has 2 equal sides -> The triangle is isosceles.
Reciprocal: If a triangle is not isosceles, then it does not have 2 equal sides.
To find the anti-reciprocal of a statement, we take the inverse of the reciprocal.
1. Anti-reciprocal of the first statement: If a quadrilateral does not have 4 equal sides, then it is not a square.
2. Anti-reciprocal of the second statement: If a triangle is not isosceles, then it does not have 2 equal sides.
Now, let's tackle the remaining statements:
3. Original statement: Only if f is integrable in an I interval, f is continuous in I.
Reciprocal: If f is not integrable in an I interval, then f is not continuous in I.
Anti-reciprocal: If f is continuous in I, then f is integrable in an I interval.
4. Original statement: If an f function is derivable at a point, then f is continuous at that point.
Reciprocal: If an f function is not derivable at a point, then f is not continuous at that point.
Anti-reciprocal: If f is continuous at a point, then f is derivable at that point.
5. Original statement: F is not an even function, so f(x) is not equal to f(-x).
Reciprocal: If F is an even function, then f(x) is equal to f(-x).
Anti-reciprocal: If f(x) is equal to f(-x), then F is an even function.
6. Original statement: If a < b, then a^2 < b^2.
Reciprocal: If a^2 < b^2, then a < b.
Anti-reciprocal: If a^2 is not less than b^2, then a is not less than b.
Remember, when finding the reciprocal or anti-reciprocal of a statement, it is crucial to consider the negation of both the hypothesis and the conclusion of the original statement.