A valid deductive argument is one in which the conclusion follows logically from the premises. In this case, the premises are "All fruits contain sugar" and "Apples are fruits". The conclusion, "Therefore, apples contain sugar", follows logically from these premises.

A deductive argument is based on a general statement (the premise) and uses logic to reach a conclusion. An inductive argument is based on a specific statement (the evidence) and uses logic to reach a general conclusion.

A mathematical proof is a logical argument that demonstrates the truth of a mathematical statement. The Pythagorean theorem can be proven using the Pythagorean triple (3, 4, 5) because it can be shown that the square of the hypotenuse (5) is equal to the sum of the squares of the other two sides (3 and 4).

A direct proof proves a statement by showing that it is true. An indirect proof proves a statement by showing that its negation is false. For example, the statement "All dogs are mammals" can be proven directly by showing that every dog is a mammal. It can also be proven indirectly by showing that the negation of the statement, "Some dogs are not mammals", is false.

The sum of the interior angles of a triangle can be proven indirectly by showing that its negation, "The sum of the interior angles of a triangle is not 180 degrees", is false. This can be done by showing that the sum of the interior angles of a triangle is always greater than 180 degrees or less than 180 degrees, which is a contradiction.

A proof by contradiction proves a statement by showing that its negation is false. A proof by cases proves a statement by considering all possible cases. For example, the statement "Every integer is either even or odd" can be proven by contradiction by showing that the negation of the statement, "There exists an integer that is neither even nor odd", is false. It can also be proven by cases by considering all possible cases: an integer is either even or odd.

The statement "Every integer is either even or odd" can be proven by cases by considering all possible cases: an integer is either even or odd. There are no other possibilities.

A constructive proof finds a solution to a problem. A non-constructive proof does not find a solution to a problem. For example, the statement "There exists a real number between 0 and 1" can be proven constructively by finding a real number between 0 and 1, such as 0.5. It can also be proven non-constructively by showing that the negation of the statement, "There does not exist a real number between 0 and 1", is false.

The statement "There exists a real number between 0 and 1" can be proven non-constructively by showing that the negation of the statement, "There does not exist a real number between 0 and 1", is false. This can be done by showing that the interval [0, 1] is non-empty.

A direct proof proves a statement for a specific case. A proof by mathematical induction proves a statement for all natural numbers. For example, the statement "The sum of the first n natural numbers is n(n+1)/2" can be proven directly by showing that it is true for a specific case, such as n = 5. It can also be proven by mathematical induction by showing that it is true for n = 1 and that if it is true for n = k, then it is also true for n = k+1.

The statement "The sum of the first n natural numbers is n(n+1)/2" can be proven by mathematical induction by showing that it is true for n = 1 and that if it is true for n = k, then it is also true for n = k+1.

A proof by exhaustion proves a statement by considering all possible cases. A proof by contradiction proves a statement by showing that its negation is false. For example, the statement "There are only a finite number of prime numbers" can be proven by exhaustion by showing that there are only a finite number of prime numbers less than any given number. It can also be proven by contradiction by showing that the negation of the statement, "There are an infinite number of prime numbers", is false.

The statement "There are only a finite number of prime numbers" can be proven by exhaustion by showing that there are only a finite number of prime numbers less than any given number.

A proof by cases proves a statement by considering all possible cases. A proof by contradiction proves a statement by showing that its negation is false. For example, the statement "Every integer is either even or odd" can be proven by cases by considering all possible cases: an integer is either even or odd. It can also be proven by contradiction by showing that the negation of the statement, "There exists an integer that is neither even nor odd", is false.

The statement "Every integer is either even or odd" can be proven by cases by considering all possible cases: an integer is either even or odd. There are no other possibilities.