Vilu c12 Math.pdf

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Page 2 :
1, , HIGHER SECONDARY – SECOND YEAR, , MATHEMATICS, ------------------------------------------, , MINIMUM LEARNING MATERIAL, (2021-2022), , CHIEF EDUCATIONAL OFFICER, VILLUPURAM DISTRICT

Page 33 :
32, , P, , q, , ∼𝒑, , ∼𝒑 ∨𝒒, , T, , T, , F, , T, , T, , F, , F, , F, , F, , T, , T, , T, , F, , F, , T, , T, , p  q  (~ p )  q, 2)Show that p q  ( p  q )  ( q  p ), P, , q, , 𝑷⟷𝒒, , T, , T, , T, , T, , F, , F, , F, , T, , F, , F, , F, , T, , P, , q, , 𝑷⟶𝒒, , 𝒒⟶𝒑, , 𝑷⟶𝒒 ∧ 𝒒⟶𝒑, , T, , T, , T, , T, , T, , T, , F, , F, , T, , F, , F, , T, , T, , F, , F, , F, , F, , T, , T, , T, , p q  (p  q)(q  p), 3)Show that pq p qis tautology., P, , q, , 𝒑∧𝒒, , 𝒑∨𝒒, , 𝒑∧𝒒 ⟶𝒑∨𝒒, , T, , T, , T, , T, , T, , T, , F, , F, , T, , T

Page 34 :
33, , F, , T, , F, , T, , T, , F, , F, , F, , F, , T, , pq p qis tautology., , 4)Show that𝑞 → 𝑝 ≡ ¬𝑝 → ¬𝑞, P, , q, , 𝑞→𝑝, , ¬𝑝, , ¬𝑞, , ¬𝑝 → ¬𝑞, , T, , T, , T, , F, , F, , T, , T, , F, , T, , F, , T, , T, , F, , T, , F, , T, , F, , F, , F, , F, , T, , T, , T, , T, , 5)Verify whether the compound statement (p  q )  [¬ (p  q ) ]is a tautology or contradiction, or contingency., P, , q, , 𝒑𝒒, , 𝒑∨𝒒, , ¬ 𝒑∨𝒒, , 𝒑𝒒 ∨ ¬ 𝒑 ∨ 𝒒, , T, , T, , T, , T, , F, , F, , T, , F, , F, , T, , F, , F, , F, , T, , F, , T, , F, , F, , F, , F, , F, , F, , T, , F, , (p q ) [¬ (p  q ) ]is contradiction., 6)Show that ¬(p q ) ≡ ( ¬p)( ¬q ), p, , q, , 𝒑𝒒, , ¬ 𝒑𝒒

Page 35 :
34, , T, , T, , T, , F, , T, , F, , F, , T, , F, , T, , F, , T, , F, , F, , F, , T, , p, , q, , ¬𝒑, , ¬𝒒, , ¬𝒑  ¬𝒒, , T, , T, , F, , F, , F, , T, , F, , F, , T, , T, , F, , T, , T, , F, , T, , F, , F, , T, , T, , T, , 7)Show that :¬ (p  q)p¬q, P, , q, , 𝑷⟶𝒒, , ¬ (p  q), , T, , T, , T, , F, , T, , F, , F, , T, , F, , T, , T, , F, , F, , F, , T, , F, , P, , q, , ∼𝒒, , T, , T, , F, , T, , T, , F, , T, , F, , p¬q

Page 39 :
38, , 6)Verify (i) Closure property (ii) associative property (iii) existence of identity (iv) existence, , of inverse and (v) commutative property for the operation ×𝟏𝟏 on a subset 𝑨 =, 𝟏, 𝟑, 𝟒, 𝟓, 𝟗 of the set of remainders 𝟎, 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, 𝟗, 𝟏𝟎, 𝑨=, , Sol:, , i), ii), iii), iv), v), , 𝟏, 𝟑, 𝟒, 𝟓, 𝟗, , 𝑿𝟏𝟏, , [1], , [3], , [4], , [5], , [9], , [1], , [1], , [3], , [4], , [5], , [9], , [3], , [3], , [9], , [1], , [4], , [5], , [4], , [4], , [1], , [5], , [9], , [3], , [5], , [5], , [4], , [9], , [3], , [1], , [9], , [9], , [5], , [3], , [1], , [4], , Elements of the table are in the elements of𝑿𝟏𝟏,Closure property is true., From the table,𝑿𝟏𝟏 is associative., From the table, the entries are symmetrical about the main diagonal.𝑿𝟏𝟏 is, commutative., The identityelement is[1]., The inverseof[1] is[1], that of [3] is[4],that of [4] is[3], that of [5] is[9], that of [9], is[5]

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