s1 - MATH 3283W. Sequences, Series, and Foundations:...

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Unformatted text preview: MATH 3283W. Sequences, Series, and Foundations: Writing Intensive. Spring 2009 Homework 1. Problems and Solutions I. Writing Intensive Part 1 (5 points). Check whether or not each of the following statements can be true for some values (”true” or ”false”) of P and Q . Write out the truth table for each of these statements. A = ( P = ⇒ Q )&( P = ⇒ ¬ Q ) , B = ( P = ⇒ Q )&( ¬ P = ⇒ Q ) , C = ( P = ⇒ Q )&( ¬ P = ⇒ ¬ Q ) . Solution. One can compose the truth tables for A,B,C by direct con- sideration of four possible cases: ( P,Q ) = ( T,T ) , ( T,F ) , ( F,T ) , and ( F,F ) . Here the equality ( P = ⇒ Q ) = ( ¬ P ∨ Q ) may be helpful. Some simplifica- tions are possible: (i) If P = T, then one of implications P = ⇒ Q or P = ⇒ ¬ Q is false, because either Q or ¬ Q is false. Therefore, in this case A = F . If P = F , then both implications are true (footnote 7 on p.2 in the textbook), so that A = T. This means that we always have A = ¬ P . (ii) If Q = T, then both implications are true, and B = T . If Q = F, then one of implications is false, and B = F. In any case, B = Q. (iii) Since ( ¬ P = ⇒ ¬ Q ) is equivalent to ( Q = ⇒ P ) (p.5 in the textbook), we have C = ( P ⇐⇒ Q ) ....
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s1 - MATH 3283W. Sequences, Series, and Foundations:...

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