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Logic and Proposition Simplification: A Study of Truth Statements, Symbolic Representation, and Proof Techniques

This assignment focuses on logic, symbolic representation, and proof techniques in mathematics.

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    Logic and Proposition Simplification: A Study of Truth Statements, Symbolic Representation, and Proof Techniques 1 . Which o f the following are statements? 1. She is a mathematics major. 2. 128=2 6 3. All that glitters is not gold. 4. Sleep tight and don’t let the bedbugs bite. Soln. A sentence that can be judged to be true or false is called a statement. 1. She is a mathematics major is a statement as it can be either true or false. 2. 128=2 6 is a statement as it is false . 3. All that glitters is not gold is a statement as it can be either true or false. 4. Sleep tight and don’t let the bedbugs bite is not a statement. 2 . Let A, B, and C be the following statements: A: John is healthy B: John is wealthy C: John is wise Use A, B, and C as defined above to translate the following statements into symbolic form. 1. John is not wealthy but he is healthy and wise. (B ′ ) ∧ (A ∧ C) 2. John is neither healthy, wealthy, nor wise. A ′ ∧ B ′ ∧ C ′ 3. John is wealthy, but he is not both healthy and wise. B ∧ ( A ∧ C ) ′ 3 . Simplify the following proposition to 2 logic operations using the laws of the algebra of propositions. Write each step on a separate line with the algebra law you used as a justification. M issing steps will be penalized. ( P ′ ∧ Q ′ ) ∨ ( P ′ ∧ Q ) ∨ ( P ∧ Q ′ ) Soln. By identity P = ( P ∨ P ) and associativity we have, ( P ′ ∧ Q ′ ) ∨ ( P ′ ∧ Q ) ∨ ( P ′ ∧ Q ′ ) ∨ ( P ∧ Q ′ ) By distributivity we have, ( P ′ ∧ ( Q ′ ∨ Q )) ∨ (( P ′ ∨ P ) ∧ Q ′ ) Reductio n by the law of excluded middle P ′ ∨ P = 1 we get, ( P ′ ∧ 1) ∨ (1 ∧ Q ′ ) B y the neutral element definition P ′ ∧ 1 = P we get,

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