Final answer:
A direct proof of a conditional statement p → q shows that if p is true, then q must also be true. A proof by contraposition shows that if q is false, then p must also be false. A proof by contradiction assumes the negation of q and demonstrates a contradiction to prove that q is true.
Step-by-step explanation:
A direct proof of a conditional statement p → q involves showing that if p is true, then q must also be true. This is done by providing logical steps and reasoning to establish the truth of q based on the truth of p. For example, in the statement 'If a number is divisible by 6, then it is divisible by 2 and 3', a direct proof would involve demonstrating that if a number is divisible by 6, it can be expressed as the product of 2 and 3, thus proving it is divisible by 2 and 3.
A proof by contraposition of a conditional statement p → q involves showing that if q is false, then p must also be false. This is done by negating both p and q in the conditional statement and demonstrating the logical connection between their negations. For example, in the statement 'If a triangle has three equal sides, then it is an equilateral triangle', a proof by contraposition would involve showing that if a triangle is not an equilateral triangle, then it cannot have three equal sides.
A proof by contradiction of a conditional statement p → q involves assuming the negation of q and demonstrating that it leads to a contradiction or an absurdity. This contradiction then implies that q must be true. For example, in the statement 'If a number is a perfect square, then it is a positive integer', a proof by contradiction would involve assuming that a perfect square is not a positive integer, which leads to a contradiction as a perfect square is defined as a positive integer multiplied by itself.
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