Question

math 118: formative assessment 2 — logic (ch. 3.4 - 3.6)
mathematical ideas (15th ed.), miller & heeren
points: 20
instructions. work individually. show clear reasoning for full credit. when asked to test validity, you may use a truth - table or name a recognized valid form. use symbols: ¬,∧,∨,→,↔.
q1. (conditionals and related statements) let (p:“x > 2”) and (q:“x^{2}>4”). (4 pts)
a) write the converse, inverse, and contrapositive of (p
ightarrow q).
b) state which statements are logically equivalent. indicate which condition is sufficient and which is necessary.
q2. (identify argument forms and validity) for each argument, circle valid or invalid and, if valid, name the form. if invalid, name the fallacy. (5 pts)
a) if the train is late, then i will miss my meeting. the train is late. therefore, i will miss my meeting. valid / invalid: ____ form/fallacy: ____
b) if it is a weekend, then the library closes early. the library closes early. therefore, it is a weekend. valid / invalid: ____ form/fallacy: ____
c) either the network is down or the password is wrong. the network is not down. therefore, the password is wrong. valid / invalid: ____ form/fallacy: ____
q3. (euler diagram reasoning) determine whether the conclusion must be true. briefly justify (an euler diagram sketch is recommended). (4 pts)
a) all artists are imaginative. some designers are artists. therefore, some designers are imaginative.
b) no managers are interns. some interns are students. therefore, no managers are students.
q4. (truth - table validity test) test each argument for validity. show key rows or a full table. (4 pts)
a) ((p
ightarrow q)wedge(q
ightarrow r),p\therefore r)
b) ((p
ightarrow q)wedge(
eg p
ightarrow r)\therefore qvee r
q5. (diagnose and repair) consider: “if a shape is a square, then it has four sides. this shape

Explanation:

Q1

Step1: Recall definitions

The converse of $p
ightarrow q$ is $q
ightarrow p$, the inverse is $
eg p
ightarrow
eg q$, and the contra - positive is $
eg q
ightarrow
eg p$. Given $p: x > 2$ and $q:x^{2}>4$.
Converse: If $x^{2}>4$, then $x > 2$.
Inverse: If $x\leq2$, then $x^{2}\leq4$.
Contra - positive: If $x^{2}\leq4$, then $x\leq2$.

Step2: Determine logical equivalence

A conditional statement $p
ightarrow q$ is logically equivalent to its contra - positive $
eg q
ightarrow
eg p$. The condition $p$ is sufficient for $q$ (if $p$ then $q$), and $q$ is necessary for $p$.

Step1: Analyze a)

The argument "If the train is late, then I will miss my meeting. The train is late. Therefore, I will miss my meeting" is of the form $p
ightarrow q,p\therefore q$, which is Modus Ponens and is valid.
Valid / Invalid: Valid
Form/Fallacy: Modus Ponens

Step2: Analyze b)

The argument "If it is a weekend, then the library closes early. The library closes early. Therefore, it is a weekend" is of the form $p
ightarrow q,q\therefore p$, which is the Fallacy of Affirming the Consequent and is invalid.
Valid / Invalid: Invalid
Form/Fallacy: Fallacy of Affirming the Consequent

Step3: Analyze c)

The argument "Either the network is down or the password is wrong. The network is not down. Therefore, the password is wrong" is of the form $p\vee q,
eg p\therefore q$, which is Disjunctive Syllogism and is valid.
Valid / Invalid: Valid
Form/Fallacy: Disjunctive Syllogism

Step1: Analyze a)

Draw an Euler diagram. The set of artists is a subset of the set of imaginative people. Since some designers are in the set of artists, some designers must be in the set of imaginative people. The conclusion is true.

Step2: Analyze b)

Draw an Euler diagram. Just because no managers are interns and some interns are students, we cannot conclude that no managers are students. The conclusion is not necessarily true.

Answer:

a) Converse: If $x^{2}>4$, then $x > 2$; Inverse: If $x\leq2$, then $x^{2}\leq4$; Contra - positive: If $x^{2}\leq4$, then $x\leq2$.
b) $p
ightarrow q$ is logically equivalent to $
eg q
ightarrow
eg p$. $p$ is sufficient for $q$, $q$ is necessary for $p$.

Q2