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s1 - MATH 3283W Sequences Series and Foundations Writing...

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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 ) . 2 (5 points). Prove that a subset of N is bounded if and only if it is finite.
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