Module 4 Homework Problems
For problems (1-7), let \(p\) be the statement “Jogging is fun” and let \(q\) be the statement “Exercise is lovely.” Construct the following sentences from the symbolic statements given.
- \(p\vee q\)
- \(\neg p \wedge q\)
- \(p \wedge \neg q\)
- \(q \vee \neg q\)
- \(p \wedge \neg p\)
- \(q\rightarrow p\)
- \(\neg p \rightarrow \neg q\)
For problems (8-14), let \(A=\{2,3,5,7,11\}\). Determine whether the following statements are true or false and explain why you chose your answer.
- \(4\in A\) or \(7\in A\)
- \(3\in A\) and \(7>1\)
- \(\{2,3,5\}\subseteq A\) and \(6\vert 360\).
- \(7\in A\) implies \(7 \notin A\) (Hint: think about when implications are true and false)
- \(7\notin A\) implies \(7\in A\)
- \(2\vert 360\) and \(3\vert 360\) implies \(6\vert 360\) (Don’t think about the meaning necessarily. Just think about truth values of each part of the statement)
- \(5\vert 321\) and \(3\vert 321\) implies \(7\vert 321\) (Yes, this answer will feel weird, but it makes sense if you think about it long enough)
Truth Tables and Logical Equivalence
- Generate a truth table for the statement \((p\vee \neg q)\wedge r\), where \(p,q,r\) are statements
- Prove the first of De Morgan’s Laws: \(\neg (p\wedge q)\equiv \neg p \vee \neg q\), where \(p,q\) are statements.
- Prove or disprove the following equivalence: \((p\wedge q) \rightarrow r \equiv (p\rightarrow r)\vee (q\rightarrow r)\)
Find the negation of the following statements.
- “If one sneezes without a tissue, then they take matters into their own hands”
- “If you cut yourself while shaving, then you have lost face.”
- “I run fast but I don’t get anywhere”
- “You are not gifted or you are not hard-working”
- “\(3\) is odd, and \(2\) is even”
- “If \(2\vert 10\) then \(2\vert 5\) or \(2\vert 7\)” (be careful with the OR when negating the consequent)
Find the converse, inverse, and contrapositive of the following statements
- “If you get 1% better every day, then you improve by 37 times in a year”
- “If you focus on positive things, then your life will be positive”
- “If \(5>4\) then \(5\leq 6\)”
Convert the following phrases from their “Necessary/Sufficient” form to their “If… then/only if/ if and only if” form.
- “Showering daily is necessary for having close friends”
- “Patience and hard work are sufficient for producing good results in life”
- “To wake up early, it is necessary and sufficient to go to bed early.”
Convert the following phrases from their “if… then/ only if/ if and only if” form to their necessary and/or sufficient form.
- “If I look up symptoms on WebMD, then I have cancer.”
- “I will thoroughly clean my house if, and only if I am having company over”
- “I will buy three years worth of toilet paper only if there’s a pandemic.”
- “\(20\vert 6000\) if and only if \(2\vert 6000\) and \(10\vert 6000\)”