Sample term test 1
- Page ID
- 243060
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Let
\[
p:\text{ The integer \(n\) is even},
\]
and
\[
q:\text{ The integer \(n\) is divisible by \(4\)}.
\]
- Write the statement \(q\rightarrow p\) in words.
- Write the inverse of \(q\rightarrow p\) symbolically.
- Write the contrapositive of \(q\rightarrow p\) in words.
- Is the converse of \(q\rightarrow p\) true?
Give a counterexample if it is false.
- Answer
-
-
Negate each of the following statements.
For parts (b) and (c), write the negation both symbolically and as an English sentence.
- \(\forall x\,\exists y\,(x>y+3).\)
- You will either finish your project, or you will not meet the deadline.
- The number \(n\) is positive and even.
- Answer
-
(a)
Construct a complete truth table for
\[
(p\vee q)\rightarrow \neg p.
\]
Then classify the proposition as a tautology,
contradiction, or neither (contingency).
- Answer
-
1.
Show that
\[
\neg(p\rightarrow q)
\]
is logically equivalent to
\[
p\wedge\neg q.
\]
You may use known logical equivalences, but each major step must be justified.
Determine whether each statement is true or false. If false, give a counterexample.
- \[
\forall x \in \mathbb{R} ,\quad x^2\geq 0.
\] - \[
\forall x \in \mathbb{R},\quad x^2>x.
\] - \[
\exists x, \in \mathbb{R}\quad x^2=x.
\] - \[
\forall x \in \mathbb{R},\quad |x|\geq x.
\]
- Answer
-
a)