Direct Proof Basics: From Hypothesis to Conclusion

In a direct proof of a conditional statement "If P, then Q", the goal is to show that P implies Q. The first step is to assume that P is true. Then, using logical deductions, definitions, and known facts, we derive that Q must be true. This approach directly establishes the implication.

For a direct proof, we start by assuming the hypothesis. Here, the hypothesis is that n is even. So, we assume n is even, meaning n = 2k for some integer k. Then, we compute n² = (2k)² = 4k² = 2(2k²), which is even, thus deriving the conclusion.

A direct proof involves starting with the given hypotheses or premises and then using logical steps, definitions, and theorems to directly arrive at the conclusion. This method does not involve assuming the negation or using contrapositive, but rather builds a chain of reasoning from the assumptions to the conclusion.

In a direct proof, we assume the hypothesis. Here, the hypothesis is x > 0. Since x is positive, multiplying both sides by x (which is positive) gives x² > 0, thus directly proving the statement.

A direct proof relies on logical implications from the hypothesis to the conclusion and does not use contradiction. Using contradiction is a feature of proof by contradiction, where we assume the negation of the conclusion and derive a false statement.

In a direct proof, we assume the hypothesis. Here, the hypothesis is that the number is divisible by 4. From this, we can show that since 4 is a multiple of 2, the number is also divisible by 2, meaning it is even.

The first step in a direct proof is to assume the hypothesis. Here, the hypothesis is that n is odd. Therefore, we begin with "Assume n is odd." Then, since n is odd, n+1 is even, which can be shown directly.

Direct proof is the technique where we assume P is true and then through logical steps derive that Q is true. This directly establishes the implication P → Q. Other techniques like contradiction or contrapositive do not involve assuming P and deriving Q directly.

In a direct proof, we start with the hypothesis that x and y are even. This means x = 2a and y = 2b for some integers a and b. Then, x+y = 2a + 2b = 2(a+b), which is even, thus proving the statement.

The standard first line in a direct proof of "If A, then B" is to assume that A is true. This sets up the hypothesis, and then we proceed to show that B must be true under this assumption.

For a direct proof, we assume the hypothesis, which is that n is divisible by 6. Since 6 is divisible by 3, it follows that n is also divisible by 3, thus proving the conclusion.

This is a direct proof of the transitive property of equality. We start by assuming the hypothesis, which is that a = b and b = c. Then, by substitution, we can conclude that a = c.

Direct proofs are well-suited for conditional statements of the form "If P, then Q" where we can assume P and derive Q through logical steps. The statement "If n is even, then n² is even" is such a conditional statement and can be proven directly by assuming n is even and showing n² is even.

In a direct proof, we assume the hypothesis that x and y are rational. We use the definition of rational numbers, this means x = a/b and y = c/d for some integers a,b,c,d with b,d ≠ 0. Then, x+y = (ad + bc)/(bd), which is rational, thus proving the statement.

We start by assuming the hypothesis that n is a multiple of 4, which means n = 4k for some integer k. Then, n² = (4k)² = 16k², which is a multiple of 16, thus proving the conclusion.