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Unformatted text preview: MATH 135 Algebra, Solutions to Assignment 2 1: (a) Make a truth table for the statement ( P â†” Â¬ R ) âˆ§ ( R â†’ Q ). Solution: Here is a truth table. P Q R Â¬ R P â†” Â¬ R R â†’ Q ( P â†” Â¬ R ) âˆ§ ( R â†’ Q ) T T T F F T F T T F T T T T T F T F F F F T F F T T T T F T T F T T T F T F T F T F F F T F T F F F F F T F T F (b) Determine whether ( P âˆ§ Â¬ Q ) âˆ¨ ( R â†” P ) is equivalent to Q â†” R . Solution: We make a truth table for ( P âˆ§ Â¬ Q ) âˆ¨ ( R â†” P ) and Q â†” R . P Q R Â¬ Q P âˆ§ Â¬ Q R â†” P ( P âˆ§ Â¬ Q ) âˆ¨ ( R â†” P ) Q â†” R T T T F F T T T T T F F F F F F T F T T T T T F T F F T T F T T F T T F F F F T F T F F F T T F F F T T F F F F F F F T F T T T Since the final two columns are not identical, the two statements are not equivalent (for example, as seen on the third row, when P is true, Q is false and R is true, the statement ( P âˆ§ Â¬ Q ) âˆ¨ ( R â†” P ) is true but the statement Q â†” R is false). (c) Determine whether P â†’ ( Q â†’ R ) is equivalent to ( P â†’ Q ) â†’ R . Solution: We make a truth table for the two given statements. P Q R Q â†’ R P â†’ ( Q â†’ R ) P â†’ Q ( P â†’ Q ) â†’ R T T T T T T T T T F F F T F T F T T T F T T F F T T F T F T T T T T T F T F F T T F F F T T T T T F F F T T T F Since the column for P â†’ ( Q â†’ R ) is not identical to the column for ( P â†’ Q ) â†’ R , these two statements are not equivalent (for example, as seen on the 6 th row, when P is false, Q is true and R is false, the statement P â†’ ( Q â†’ R ) is true but the statement ( P â†’ Q )...
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 Fall '08
 ANDREWCHILDS
 Logic, Algebra, R. Indeed

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