Download Direct and Converse Theorems. The Elements of Symbolic Logic by I. S. Gradshtein, I. N. Sneddon, M. Stark, S. Ulam PDF

By I. S. Gradshtein, I. N. Sneddon, M. Stark, S. Ulam

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They p o s s e s s t h e p r o p e r t y This theorem is obtained from the first theorem by substituting for each condition and conclusion its negation and is known as the inverse (or contrary) theorem to the first theorem. e. if the converse theorem is true then the inverse theorem is also true, and, conversely, if the inverse theorem is true then 41 Direct and converse theorems so is the converse*. The i n v e r s e theorem t o the c o n v e r s e i s formulated t h u s : I f t h e elements o f the set_M db n o t p o s s e s s the p r o p e r t y P ( p o s s e s s the p r o p e r t y (3 ) then n e i t h e r do they p o s s e s s the p r o p e r t y a ( t h e y p o s s e s s the p r o p e r t y a ) , and t h i s means t h a t t h e e l e m e n t s o f the s e t M p o s s e s s i n g the p r o p e r t y a, must n e c e s s a r i l y p o s s e s s the p r o p e r t y p, i n o t h e r words, t h a t the o r i g i n a l ( d i r e c t ) theorem i s v a l i d .

To t h e e l e m e n t s o f the s e t M n o t p o s s e s s i n g p r o p e r t y |3 c o r r e s p o n d s t h e r e g i o n BCDM^K o f o u r diagram. This r e g i o n , as i t l i e s o u t s i d e t h e r e c t a n g l e AK L^M^ i s n o t hatched and t h e r e f o r e i n v i r t u e o f t h e d i r e c t theorem t h e r e a r e no s i t e s c o l o u r e d grey i n i t , i . e . , i f an element o f t h e s e t M d o e s n o t p o s s e s s the p r o p e r t y p ( o f b e i n g h a t c h e d ) , then n e i t h e r d o e s i t p o s s e s s t h e p r o p e r t y a ( o f b e i n g grey).

39 Direct and converse theorems Guided by t h e theorems o f geometry we have drawn a diagram showing t h e c o n n e x i o n between t h e p r o p e r t i e s o f the q u a d r i l a t e r a l s t h a t we a r e i n v e s t i g a t i n g , The diagram g r a p h i c a l l y shows us t h e s e p r o p e r t i e s , and by l o o k i n g a t i t we can f o r m u l a t e t h e c o r r e s p o n d i n g t h e o r ems. The v a l i d i t y o f t h e s e theorems, n a t u r a l l y , depends upon the v a l i d i t y o f o u r diagram.

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