Question

math 118: formative assessment 2 — logic (ch. 3.4 - 3.6) points: 20 mathematical ideas (15th ed.), miller & heeren 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: “x² > 4”. (4 pts) a) write the converse, inverse, and contrapositive of p → 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 → q)∧(q → r), p ∴ r b) (p → q)∧(¬p → r) ∴ q∨r q5. (diagnose and repair) consider: “if a shape is a square, then it has four sides. this shape has four sides. therefore, this shape is a square.” (3 pts)

Explanation:

Q1.

a)

- **Converse**:
The conditional is \(p ightarrow q\) where \(p: x > 2\) and \(q:x^{2}>4\). The converse of \(p ightarrow q\) is \(q ightarrow p\), so it is "If \(x^{2}>4\), then \(x > 2\)".
- **Inverse**:
The inverse of \(p ightarrow q\) is \( eg p ightarrow eg q\). So it is "If \(x\leq2\), then \(x^{2}\leq4\)".
- **Contra - positive**:
The contra - positive of \(p ightarrow q\) is \( eg q ightarrow eg p\). So it is "If \(x^{2}\leq4\), then \(x\leq2\)".

b)

- **Logical equivalence**:
The original conditional \(p ightarrow q\) and its contra - positive \( eg q ightarrow eg p\) are logically equivalent. The inverse \( eg p ightarrow eg q\) and the converse \(q ightarrow p\) are logically equivalent to each other.
- **Sufficient and necessary conditions**:
The condition \(p\) is sufficient for \(q\) (if \(x > 2\), then \(x^{2}>4\)). The condition \(q\) is necessary for \(p\) (if \(x^{2}\leq4\), then \(x\leq2\), so for \(x > 2\) to be true, \(x^{2}>4\) must be true).

Q2.

a)

- **Validity and form**:
This is a valid argument. The form is Modus Ponens. Given \(p ightarrow q\) (If the train is late, then I will miss my meeting) and \(p\) (The train is late), we can conclude \(q\) (I will miss my meeting). So, Valid / Modus Ponens.

b)

- **Validity and form**:
This is an invalid argument. The fallacy is Affirming the Consequent. Given \(p ightarrow q\) (If it is a weekend, then the library closes early) and \(q\) (The library closes early), we wrongly conclude \(p\) (it is a weekend). So, Invalid / Affirming the Consequent.

c)

- **Validity and form**:
This is a valid argument. The form is Disjunctive Syllogism. Given \(p\vee q\) (Either the network is down or the password is wrong) and \( eg p\) (The network is not down), we can conclude \(q\) (the password is wrong). So, Valid / Disjunctive Syllogism.

Q3.

a)

- **Conclusion and justification**:
The conclusion "Some designers are imaginative" must be true. In an Euler - diagram, the set of artists is a subset of the set of imaginative people, and the set of some designers is a subset of the set of artists. So, the set of some designers must be a subset of the set of imaginative people.

b)

- **Conclusion and justification**:
The conclusion "No managers are students" is not necessarily true. In an Euler - diagram, the set of managers and interns are disjoint, but the set of interns and students overlap. There is no information to suggest that the set of managers and students are disjoint.

Q4.

a)

- **Truth - table construction**:
| \(p\) | \(q\) | \(r\) | \(p ightarrow q\) | \(q ightarrow r\) | \((p ightarrow q)\wedge(q ightarrow r)\) | \((p ightarrow q)\wedge(q ightarrow r)\wedge p\) | \(r\) |
|---|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T | T |
| T | T | F | T | F | F | F | F |
| T | F | T | F | T | F | F | T |
| T | F | F | F | T | F | F | F |
| F | T | T | T | T | T | F | T |
| F | T | F | T | F | F | F | F |
| F | F | T | T | T | T | F | T |
| F | F | F | T | T | T | F | F |
The argument \((p ightarrow q)\wedge(q ightarrow r),p\therefore r\) is valid since whenever \((p ightarrow q)\wedge(q ightarrow r)\) and \(p\) are true, \(r\) is true.

b)

| \(p\) | \(q\) | \(r\) | \(p ightarrow q\) | \( eg p ightarrow r\) | \((p ightarrow q)\wedge( eg p ightarrow r)\) | \(q\vee r\) |
|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T |
| T | T | F | T | F | F | T |
| T | F | T | F | T | F | T |
| T | F | F | F | F | F | F |
| F | T | T | T | T | T | T |
| F | T | F | T | T | T | T |
| F | F | T | T | T | T | T |
| F | F | F | T | F | F | F |
The argument \((p ightarrow q)\wedge( eg p ightarrow r)\therefore q\vee r\) is valid since whenever \((p ightarrow q)\wedge( eg p ightarrow r)\) is true, \(q\vee r\) is true.

Q5.

- **Fallacy identification**:
This is an example of Affirming the Consequent. Given \(p ightarrow q\) (If a shape is a square, then it has four sides) and \(q\) (This shape has four sides), we wrongly conclude \(p\) (this shape is a square).

Answer:

Q1.

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) \(p
ightarrow q\) and \(
eg q
ightarrow
eg p\) are logically equivalent; \(
eg p
ightarrow
eg q\) and \(q
ightarrow p\) are logically equivalent. \(p\) is sufficient for \(q\), \(q\) is necessary for \(p\)

Q2.

a) Valid / Modus Ponens
b) Invalid / Affirming the Consequent
c) Valid / Disjunctive Syllogism

Q3.

a) The conclusion must be true.
b) The conclusion is not necessarily true.

Q4.

a) The argument is valid.
b) The argument is valid.

Q5.

The fallacy is Affirming the Consequent.