74hwsol2 - Solutions to Homework Assignment 2 Math 74 Fall...

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Solutions to Homework Assignment 2 Math 74, Fall 2006 October 1, 2006 1. (a) If P and Q are statements, then using De Morgan’s laws we see that ¬ [ ¬ P ∧ ¬ Q ] ⇔ ¬¬ P ∨ ¬¬ Q P Q. Likewise, ¬ [ P ∧ ¬ Q ] P Q, and ( ¬ [ P ∧ ¬ Q ]) ( ¬ [ Q ∧ ¬ P ]) ( P Q ) . So we see that we can build all of the logical connectives out of ¬ and . (b) Let P , Q and R be statements. Then using our work above we have [( P Q ) R ] [ ¬ Q ( R P )] ¬ ([( P Q ) R ] ∧ ¬ [ ¬ Q ( R P )]) ¬ ([ ¬ ( P ∧ ¬ Q ) R ] [ Q ∧ ¬ ( R P )]) ¬ ( ¬ [ ¬ R P ∧ ¬ Q ] [ Q ∧ ¬ R ∧ ¬ P ]) . From this point, it is easy to see that the statement is equivalent to ¬ [ Q ∧ ¬ R ∧ ¬ P ]. 2. (a) We will show that we can write the five common logical connectives ¬ , , , , by using only the logical connective . It is clear that P P ⇔ ¬ [ P P ] ⇔ ¬ P. Using De Morgan’s laws, we see that ( P P ) ( Q Q ) ( ¬ P ) ( ¬ Q ) ⇔ ¬ [ ¬ P ∧ ¬ Q ] P Q. Also, ( P Q ) ( P Q ) ⇔ ¬ ( P Q ) ⇔ ¬ [ ¬ ( P Q )] P Q. Also, [( P P ) ( P P )] [ Q Q ] [( ¬ P ) ( ¬ P )] [ Q Q ] ( ¬ P ) Q ( P Q ) . 1
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From our work above we have that [ P Q ] [ P Q ] [ Q P ] [( P Q ) ( Q P )] [( P Q ) ( Q P )] [ R S ] [ R S ] , where the symbol R is used in place of the statement [( P P ) ( P P )] [ Q Q ] and S is used in place of [( Q Q ) ( Q Q )] [ P P ] . (b) We will show that we can write the five common logical connectives ¬ , , , , by using only the logical connective . It is clear that P P ⇔ ¬ [ P P ] ⇔ ¬ P. Using De Morgan’s laws, ( P P ) ( Q Q ) ⇔ ¬ [( ¬ P ) ( ¬ Q )] P Q. Also, ( P Q ) ( P Q ) ⇔ ¬ ( P Q ) ⇔ ¬ [ ¬ ( P Q )] P Q.
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