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# Solution - Without using truth tabic show that ~(p v q)v(~p ∧ q) = ~p - HSC Science (General) 12th Board Exam - Mathematics and Statistics

#### Question

Without using truth tabic show that ~(p v q)v(~p ∧ q) = ~p

#### Solution

~(p v q)v(~p ∧ q)

≡~(p v q)v~(p ∨ ~q)                      by De Morgan's Law

≡~[(p ∨ q) ∧ (p ∨ ~q)]                    by De Morgan's Law

≡~{[(p ∨ q) ∧ p] ∨ [(p ∨ q)∧ ~q)]}   by Distributive Law

≡ ~{[p] ∨ [(p ∨ q) ∧ ~q]}               by  Absorption Law

≡ ~{[p] ∨ [(p∧ ~q) ∨ (q ∧ ~q)]}      by Distributive Law

≡~{[p] ∨ [(p ∧ ~q) ∨ F]}                by Complement Law

≡~{[p] ∨ [(p ∧ ~q)]}                     by Identity Law

≡~p ∧ (~p ∨ q)                             by De Morgan's Law

≡ ~p                                           by Absorption Law

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#### APPEARS IN

2015-2016 (March) (with solutions)
Question 3.1.2 | 3 marks
Solution for question: Without using truth tabic show that ~(p v q)v(~p ∧ q) = ~p concept: null - Algebra of Statements. For the courses HSC Science (General) , HSC Science (Computer Science), HSC Science (Electronics), HSC Arts
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