By Smirnov V.I.
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Prove by using different laws. (a) Ø (P Ú Q) Ú (Ø P Ù Q) º Ø P (b) P Ú (P Ù Q) º P (c) (P Ú Q) Ù Ø P º Ø P Ù Q 5. Write each of the following in symbolic form by indicating statements. (a) Brown is rich and unhappy. (b) Jackson speaks English or French. (c) I am hungry and I can study. (d) I am tired if and only if I work hard. (e) If New York is a city, then it is the capital of US. (f) 5 + 2 = 7 if 7 - 2 = 5. 6. Write the truth value of each of the following statements. (a) Sun rises in the south.
I) A = {a, b, c} (ii) B = {a, e, i, o, u} (iii) C = {b, c, o, u} (iv) D = {b, c, i, o, u, k}. 16. In a class containing 100 students, 30 play tennis; 40 play cricket; 40 do athletics; 6 play tennis and cricket; 12 play cricket and do athletics; and 10 play tennis and do athletics; while 14 play no game or do athletics at all. How many play cricket, tennis and do athletics? 17. If in a city 70% of the residents can speak French and 50% can speak English, what percentage of residents can speak both the languages, if 10% residents cannot speak any of these two languages?
D| = 2 Therefore, |A| = |B| = |D| = 2 ¹ |C| = 5; So A, B and D are equivalent. Example 23 For all Sets A and B prove that (A Ç B) È (B A) = B. Solution: (A Ç B) È (B A) = (A Ç B) È (B Ç Ac) Example 24 Solution: = ((A ÇB) È B) Ç ((A Ç B) È Ac) [Distributive law] = B Ç ((A Ç B) È Ac) [Absorption law] c c = B Ç ((A È A ) Ç (B È A )) [Distributive law] = B Ç (U Ç (B È Ac)) [Complement law] = B Ç (B È Ac) =B [ Absorption law] By applying properties of sets prove that (A B) Ç (B A) = f for all sets A and B.