Sample term test 1
- Page ID
- 243060
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These mock exams are provided to help you prepare for Term/Final tests. The best way to use these practice tests is to try the problems as if you were taking the test. Please don't look at the solution until you have attempted the question(s). Only reading through the answers or studying them, will typically not be helpful in preparing since it is too easy to convince yourself that you understand them.
You can try timing yourself for \(40\) minutes.
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
-
-
If the integer \(n\) is divisible by 4, then \(n\) is even.
-
\(\neg q \rightarrow \neg p\)
-
If the integer \(n\) is not even, then \(n \) is not divisible by \(4 \).
-
The converse is \(p \rightarrow q\): If \(n\) is even, then \(n\) is divisible by \(4.\)
No, the converse is false.
Counterexample: Let \(n = 6\) . The integer \(n = 6\) is even (\(p\) is true), but \(6\) is not divisible by \(4\) (\(q\) is false).
-
Negate each of the following statements.
For parts (2) and (3), write the negation both symbolically and as an English sentence.
- \(\forall x\,\exists y\) such that \(x>y+3.\)
- You will either finish your project, or you will not meet the deadline.
- The number \(n\) is positive and even.
- Answer
-
1. \(\neg (\forall x \exists y (x > y + 3)) \equiv \exists x \forall y (x \le y + 3)\)
2. Let \(p\): You will finish your project, and \(q\): You will meet the deadline.
Original form: \(p \lor \neg q\)
Symbolic Negation: \(\neg(p \lor \neg q) \equiv \neg p \land q\)
English Sentence: You will not finish your project, and you will meet the deadline. (You will not finish your project, but you will meet the deadline.)3. Let \(p\): \(n\) is positive, and \(q\): \(n\) is even.
Original form: \(p \land q\)
Symbolic Negation:\(\neg(p \land q) \equiv \neg p \lor \neg q\)
English Sentence: The number \(n\) is not positive or \(n\) is odd (not even).
Construct a complete truth table for
\(
(p\vee q)\rightarrow \neg p.
\)
Then classify the proposition as a tautology,
contradiction, or neither (contingency).
- Answer
-
Since the truth values in the final column contain both T and F, the proposition is neither a tautology nor a contradiction; it is a contingency.
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.
- Answer
-
\begin{align*}
\neg(p \rightarrow q) &\equiv \neg(\neg p \lor q) && \text{(Conditional Law: } A \rightarrow B \equiv \neg A \lor B\text{)} \\
&\equiv \neg(\neg p) \land \neg q && \text{(De Morgan's Law)} \\
&\equiv p \land \neg q && \text{(Double Negation Law)}
\end{align*}
Thus, \(\neg(p \rightarrow q) \equiv p \land \neg q\).
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
-
1. True. The square of any real number is always non-negative.
2. False.
Counterexample:}Let \(x = 0.5, x^2 = 0.25\), and \(0.25 \ngtr 0.5\).3. True.
Proof/Example:For \(x = 1, 1^2 = 1\), which satisfies the equation.4. True. By definition of absolute value, if \(x \ge 0\), then \(\vert{}x\vert{} = x \ge x;\) if \(x < 0\), then \(\vert{}x\vert{} = -x > 0 > x.\)