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p--> q
Logically equivalent to contrapositive |
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q-->p
Logically equivalent to inverse |
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| ¬p-->¬q Logically equivalent to converse |
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| ¬q--> ¬p logically equivalent to statement |
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(a-->b)=(¬b-->¬a)
Assume ¬b, then demonstrate this assumption forces ¬a. |
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| we show that ¬(a-->b) is impossible. Assume 'not b', then show a contradiction if a^¬b were to be the statement. |
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| Assume 'a', show that it forces 'b'. |
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| non-constructive existence proof |
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| we prove some specific object exists by showing it can't not exist |
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| Proof by mathematical induction |
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to prove statement P(n)
1. verify p(0) is true
2. assume p(k) is true for some generically specified k
3. show p(k) true--> p(k+1) true |
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A relation R on a set A is called reflexive if (a,a)E R for every element aEA.
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| A relation R on a set A is symmetric if (b,a) E R whenever (a,b) E R, for all a,b E A. |
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| A relation R on a set A such that for all a,b E A if (a,b) E R and (b,a)ER, then a=b is called antisymmetric |
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A relation R on a set A is called transitive if whenever (a,b) E R and (b,c)E R, then (a,c) E R, for all a, b, c E A.
The relation R on a set A is transitive if and only if Rn C R for n=1, 2, 3,.. |
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