When proving p right arrow q using a DIRECT proof what is as

When proving p right arrow q using a DIRECT proof, what is assumed and what must be proved? Assume prove When proving p right arrow q using a proof by CONTRADICTION, what is assumed and what must be proved? Assume Prove When proving p right arrow q using an INDIRECT proof (proving the CONTRAPOSITIVE), what is assumed and what must be proved? Assume Proved. When disproving a statement of the form \"forevery x P(x)\", what must be proved?

Solution

(a) in oredr to prove that p->q, assume that p exists, and prove that q also exist with the help of p

(b) In order to prove that p->q bycontradiction method, assume that q does not hold, which leads us to contradiction

e.g. if x>0 and y>0, then x+y>0.

proof: Assume that x+y<0, then x<-y

impies that x<0, which is a contradiction as x>0, so x+y>0

(c) Proof by contrapositive takes advantage of the logical equivalence between \"P implies Q\" and \"Not Q implies Not P\". For example, the assertion \"If it is my car, then it is red\" is equivalent to \"If that car is not red, then it is not mine\". So, to prove \"If P, Then Q\" by the method of contrapositive means to prove \"If Not Q, Then Not P\"

(d) in order to disprove some statement, simply produce some counter-example

e.g. Sum of two positive numbers is negative, to disprove some statement, simply produce some counter-example, that is 2+3=5>0 not negative