I Need Help solving this practice quiz for my Computer Science Discrete Mathematics Class:

1. Using only the rules of inference and the logical equivalences listed on the last page of this quiz, show that the following argument is a contradiction by reducing it to a value of "False". You may assume that all the premises given are true. Make sure that you include both the rule and the line number(s) to which that rule is applied. [7 marks]

f

2. Using only the rules of inference and the logical equivalences listed on the last page of this quiz, show that the following argument is valid. You may assume that all the premises given are true. Make sure that you include both the rule and the line number(s) to which that rule is applied. [9 marks]

1) () ()

2) (() ()) ()

3) () ()

4) () ()

() ()

3. Assume that n and m are both integers. Prove the following:

If n is an odd number and m is an even number, then m - n is an odd number.

To complete this proof you may use algebra, the logical equivalences, the inference rules, and the following additional "rules":

"the definition(s) of even numbers" is an even number if and only if can be written as 2 for some integer x is an even number if and only if n is not an odd number

"the definition(s) of odd numbers" is an odd number if and only if can be written as 2 + 1 for some integer x is an odd number if and only if n is not an even number

Although you are not expected to refer to the algebraic laws that you use by name, you must still SHOW EVERY STEP in detail, even when simply performing algebra.

Formula Sheet:

Logical Equivalences

= } Identity { =

= } Domination { =

= } Idempotence { =

= } Negation { =

( ) = } DeMorgan's Law { ( ) =

= } Commutativity { =

( ) = ( ) } Associativity { ( ) = ( )

( ) = ( ) ( ) } Distributivity { ( ) = ( ) ( )

( ) = } Absorption { ( ) =

= } Implication Equivalence

= } Contraposition

= ( ) ( ) } Biconditional Equivalence

() = } Double Negation

Inference Rules

{ } Addition

{ } Simplification

{ } Conjunction

{ } Resolution

{ } Modus Ponens

{ } Modus Tollens {

} Disjunctive Syllogism

{ } Hypothetical Syllogism

{ () () } Universal Instantiation

{() () } Existential Generalization

{ () () } Existential Instantiation (for a "fresh" variable )

{() () } Universal Generalization (with "arbitrary" variable )