Chain closure
If under given conditions the statements "If A, then B" and "If B, then C" are true, then under these conditions the statement "If A, then C" is also true - do my php homework . Short form of the chain closure: [(A⇒B)∧(B⇒C)]⇒(A⇒C)
Example: If for every natural number a the statements "If 18 divisors of a, then 9 divisors of a" and "If 9 divisors of a, then 3 divisors of a" are true, then the statement "If 18 divisors of a, then 3 divisors of a" is also true for all a∈N.
Inference to an all statement
If for any a the statement A(a) is true, then the all-statement "For every (all) x is true " A(x) is true.
In the following we give an example of inference to an all-statement:
Example: If it is proved that the theorem of Pythagoras is true for any right triangle, then it is true for all right triangles.
Rule of contraposition
If the propositional connection "If A, then B" is true, then "If not B, then not A" is also true (and vice versa).
Short form of the rule of contraposition: (A⇒B)⇔(¬ B⇒¬ A)
Proof (with the help of the truth value table):
Figure
Example: If the statement "If two triangles are congruent to each other, they are also similar to each other" is true, then - mathxl answers : "If two triangles are not similar to each other, then they are also not congruent to each other."
Sometimes the contraposition of a proposition is easier to prove than the actual proposition - help me with my homework please . This is often the case when the converse of a proposition is to be proved (cf. the following example on the equivalence conclusion).
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