Question

1.Write the negation of the following statement in English. Do
not simply add the words “not” or “it is not the case that” or
similar before the existing statement. *“Every dog likes
some flavor of Brand XYZ dog food.”*

2. Prove that the compound statements are logically equivalent by using the basic logical equivalences (13 rules). At each step, state which basic logical equivalence you are using. (p → q) ∧ (¬r → q) and (p ∨ ¬r) → q

Answer #1

1. The negation of the given statement is "At least one dog does not like a flavor of BRAND XYZ dog food".

2. The sign '=' will be used as equivalence between statements

Conditional rule

Distributive rule

Distributive rule

Distributive rule

The second statement (in the bracket) becomes q because when q is true the statement is true and when q is false it is false.

De Morgan's rule

Consider the following (true) statement:
“All birds have wings but some birds cannot fly.”
Part 1
Write this statement symbolically as a conjunction of two
sub-statements, one of which is a conditional and the other is the
negation of a conditional.
Use three components (p, q, and r)
and explicitly state what these components correspond to in the
original statement.
Hint: Any statement in the form "some X cannot Y"
can be rewritten equivalently as “not all X can Y,”...

Discrete Math
1. Write the compound statements in disjunctive
normal form. (Since they are logically equivalent, they have the
same disjunctive normal form, so you only need to give one
answer.
(p → q) ∧ (¬r → q) and (p ∨ ¬r) → q
2. Consider the premises:
• It is not snowing today and it is windy;
• School will be canceled only if it is snowing today;
• If school is not canceled today, then our study group will...

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