(definition of conditional, second set of () unnecessary but written for clarity)
≡ (? ∧ ¬?) ∨ (¬? ∨ ?)
(de Morgan, second set of () unnecessary but written for clarity)
≡ (? ∨ ¬? ∨ ?) ∧ (¬? ∨ ¬? ∨ ?)
(distributive law)
≡ (¬? ∨ ?) ∧ (¬? ∨ ?)
(idempotent law)
≡ (¬? ∨ ?)
(idempotent law)
≡ ? → ?
(definition of conditional).
An even shorter solution is produced by just using the absorption law after de Morgan:
(? → ?) → (? → ?)
≡ ¬(¬? ∨ ?) ∨ (¬? ∨ ?)
(definition of conditional)
≡ (? ∧ ¬?) ∨ ¬? ∨ ?
(de Morgan)
≡ ((? ∧ ¬?) ∨ ¬?) ∨ ?
(technically unnecessary () introduced to show how the absorption law applies)
≡ ¬? ∨ ?
(absorption law)
≡ ? → ?
(definition of conditional

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2.
Suppose two integers x and y are given. Formulate the
contrapositive
of the following statement, which is
given in both verbal and symbolic form:
If x is positive and y is non-negative, then xy is non-negative.
? > 0 ∧ ? ≥ 0 → ?? ≥ 0
.
You must also give your answer in both forms.
The contrapositive, in verbal form:
If xy is negative, then x is non-positive, or y is negative.
The contrapositive, in symbolic form:
?? < 0 → ? ≤ 0 ∨ ? < 0
3.
Rewrite in standard
“if.. then” form:
.

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