Prove: [ P -> Q AND R -> S AND (P OR R) ] -> (Q OR S) -> NOT, ---
1. P -> Q ___ hypothesis 2. R -> S ___ hypothesis 3. P OR R ___ hypothesis 4. ~P OR Q ___ implication from hyp 1. 5. ~R OR S ___ implication from hyp 2 6. ~P OR Q OR S ___ addition to 4. 7. ~R OR Q OR S ___ addition to 5. 8. Let T == (Q OR S) ___ substitution 9. (~P OR T) AND (~R OR T) ___ Conjunction 6,7 10. T OR (~P AND ~R) ___ Distribution from 9 11. T OR ~(P OR R) ___ De Morgan's theorem 12. Let M == (P OR R) ___ substitution
13. (T OR ~M) AND M ___ conjunction 11, hyp 3 From there, you can use distribution to get (T AND M) OR (~M AND M). The contradiction goes away leaving you with T AND M, which can simplify to T.
Please use the discussion area to explain this fascinating question Constructive Dilemma Proof: 1. ( A -> B ) Premise 2. ( C -> D ) Premise 3. ( A or C ) Premise 4. ( - B -> - A ) 1 , Contraposition 5. ( - A -> C ) 3 , Implication 6. ( - B -> C ) 4 , 5 Chain Argument 7. ( - B -> D ) 6 , 2 Chain Argument 8. ( B or D ) 7 Implication , Q.E.D.
Yes. The square root of a positive integer can ONLY be either:* An integer (in this case, it isn't), OR * An irrational number. The proof is basically the same as the proof used in high school algebra, to prove that the square root of 2 is irrational.
No. The sequence (n*sin(n)) is not properly divergent. To be properly divergent it must either "tend to" +inf or -inf. We say that (xn) tends to +inf if for every real number a there exists a natural number N such that if n>=N, then xn>a. It is clear that no such N exists for all real numbers because n*sin(n) oscillates (because of the sin(n)). Therefore (n*sin(n)) is not properly divergent. This is not a rigorous proof but the definition of proper divergence is precise and can be used for any proof dealing with proper divergence.
Proof by contradiction is also known by its Latin equivalent, reductio ad absurdum.
45% The percentage of alcohol is always 1/2 of the proof.
Mathematical logic and proof theory (a branch of mathematical logic) for proof
Mathematical logic.
Axioms and logic (and previously proved theorems).
Proof
True or False. Logic and proof in math have been in existence sine the time of the ancient Greeks. true
no
In normal math, 1 is not equal to 0, so any "proof" that they are equal either uses non-standard definitions, or it is based on faulty logic.
true
Yo could try using logic.
At least three thousand years.
true
No, logic and formal proof have been integral parts of mathematics for thousands of years. The ancient Greeks, such as Euclid and Pythagoras, were known for their use of logical reasoning and formal proofs. However, the development of formal logic as a field of study did occur more recently in the 19th and 20th centuries.