What We Already Know ✅

Statements and their truth values (T/F)
Basic logical connectives (∼, ∧, ∨)
Truth tables
Logical equivalence (≡)
Logical equivalence laws

The Implication: A One-Way Logical Street 🚦

What is a Conditional Statement?

Symbol: → (arrow)

Definition: p → q means "If p, then q" or "p implies q"

In every conditional statement, we have two key parts (Unit 1.2, Slide 3):

p is called the hypothesis (or antecedent or premise) - the "if" part
q is called the conclusion (or consequent) - the "then" part

How Do We Determine If a Conditional is True?

The truth table for p → q reveals something interesting (Unit 1.2, Slide 2):

p	q	p → q	What's happening here?
T	T	T	True hypothesis leads to true conclusion (Promise kept!)
T	F	F	True hypothesis leads to false conclusion (Promise broken!)
F	T	T	False hypothesis leads to true conclusion (No promise made)
F	F	T	False hypothesis leads to false conclusion (No promise made)

Notice something crucial: the conditional p → q is false in only one case—when p is true and q is false. In all other cases, it's true!

The "Vacuously True" Cases

You might be wondering about those last two rows. Why is "If p, then q" true when p is false?

Think of a conditional as making a promise: "If this happens, then that will happen." If the "if" part never happens (p is false), then the promise isn't broken—regardless of whether q is true or false. It's like promising "If it rains today, I'll bring an umbrella." If it doesn't rain, the promise wasn't tested, so it can't be false.

This is called being vacuously true—true in a way that's empty or untested.

Some examples that might seem strange but are logically TRUE:

"If the moon is made of green cheese, then I am the King of France." (Both parts are false, but the conditional is true!)
"If 0 = 1, then 1 = 2." (Again, both false, but the conditional is true)
"If pigs can fly, then 2+2=4." (False premise, true conclusion, still a true conditional)

💭 Question: Let p be "2+2=5" (F) and q be "Pretoria is the capital of South Africa" (T). What is the truth value of p → q? Why?

(Take a moment to think about it...)

The answer is TRUE! Since p is false (2+2 is not 5), the conditional is automatically true regardless of q's value. The promise was never tested because the hypothesis didn't occur.

Many Ways to Say the Same Thing: Conditional Expressions in Natural Language 🗣️

The conditional p → q can be expressed in many ways in English, and recognizing these variations is crucial (Unit 1.2, Slide 4):

If p, then q.
p implies q.
q if p. (Notice how the order flips here!)
p only if q. (This one's tricky - see below)
q whenever p.
p is sufficient for q. (p being true is enough to guarantee q is true)
q is necessary for p. (q must be true for p to have any chance of being true)

Let's look more closely at "p only if q" since it often causes confusion:

When we say "p only if q," we mean that the ONLY situation where p can be true is if q is also true. If q is false, then p must be false. Mathematically, this is equivalent to p → q.

Example: "You can access the network only if you have a valid password."

Let p = "You can access the network"
Let q = "You have a valid password"
This means p → q
In other words, if you don't have a valid password (∼q), then you can't access the network (∼p)

💭 Practice questions:

Let r: x ≥ 1, p: x > -1. Translate "x ≥ 1 only if x > -1" into symbolic form. (Similar to Tutorial Test 1A, Q1b)
Let p: k is prime, s: k does not have last digit 0. Translate "A necessary condition for k to be prime is that k does not have last digit 0" into symbolic form. (Similar to Tutorial Test 1C, Q1c)
Let r: k is even, s: k is a multiple of 4. Translate "A sufficient condition for k to be even is that k is a multiple of 4" into symbolic form. (Similar to Tutorial Test 1F, Q1c)

(Take a moment to think about these...)

Answers:

r → p (If x ≥ 1, then x > -1)
p → s (If k is prime, then k does not have last digit 0)
s → r (If k is a multiple of 4, then k is even)

The Implication Law: A Surprising Equivalence 💡

Here's something that might surprise you (Unit 1.2, Slide 6):

p → q ≡ ∼p ∨ q

In words: "If p then q" is logically equivalent to "not p or q"!

Let's verify this with a truth table:

p	q	p → q	∼p	∼p ∨ q
T	T	T	F	T
T	F	F	F	F
F	T	T	T	T
F	F	T	T	T

The columns for p → q and ∼p ∨ q are identical, confirming they're logically equivalent.

This makes intuitive sense if you think about it: p → q is false only when p is true and q is false. Similarly, ∼p ∨ q is false only when ∼p is false (meaning p is true) AND q is false.

This law is incredibly useful because it allows us to convert conditionals into expressions using only NOT and OR, which can simplify many proofs and analyses.

When a Conditional Statement is False: The Negation 🚫

What does it mean to negate a conditional statement? Let's find out (Unit 1.2, Slide 7):

∼(p → q) ≡ p ∧ ∼q

In words: The negation of "If p then q" is "p and not q."

We can prove this easily using our Implication Law and De Morgan's Law: ∼(p → q) ≡ ∼(∼p ∨ q) (Implication Law) ≡ ∼(∼p) ∧ ∼q (De Morgan's Law) ≡ p ∧ ∼q (Double Negative Law)

Intuitively, this makes sense: The only way for "If p, then q" to be false is if p is true but q is false.

Example: The negation of "If it is raining, then the ground is wet" is "It is raining and the ground is not wet."

(Notice it's NOT "If it is raining, then the ground is not wet" or "If it is not raining, then the ground is wet.")

💭 Question: Let p: I study chemistry, q: I study IT, r: I study statistics. Write the negation of the statement form q ∧ p → r without using "It is not the case that". (Similar to Tutorial Test 1B, Q1c)

(Think about this before checking the answer!)

Answer: (q ∧ p) ∧ ∼r. In words: "I study IT and chemistry, and I do not study statistics."

Variations on a Theme: Converse, Inverse, and Contrapositive

Given a conditional statement p → q, we can form three important related statements (Unit 1.2, Slide 3):

Converse: q → p (Swap hypothesis and conclusion)
Inverse: ∼p → ∼q (Negate both hypothesis and conclusion)
Contrapositive: ∼q → ∼p (Swap and negate hypothesis and conclusion)

Let's see an example (Unit 1.2, Slide 4):

Original statement: "If n is a multiple of 6, then n is a multiple of 3." (True)

Let p = "n is a multiple of 6"
Let q = "n is a multiple of 3"

Converse: "If n is a multiple of 3, then n is a multiple of 6." (False - consider n=3) Inverse: "If n is not a multiple of 6, then n is not a multiple of 3." (False - consider n=3) Contrapositive: "If n is not a multiple of 3, then n is not a multiple of 6." (True)

💭 Question: Let the statement be "If I study IT and chemistry, then I study statistics." Write the inverse of this statement. (Similar to Tutorial Test 1B, Q1b)

Answer: "If it is not the case that I study both IT and chemistry, then I do not study statistics." Or equivalently: "If I do not study IT or I do not study chemistry, then I do not study statistics."

Important Relationships Between These Forms 🔑

Now, let's explore some crucial relationships (Unit 1.2, Slides 5-7):

Converse Error and Inverse Error

A common logical fallacy is assuming that if a conditional statement is true, its converse or inverse must also be true. This is incorrect!

p → q is NOT logically equivalent to its converse q → p. p → q is NOT logically equivalent to its inverse ∼p → ∼q.

Our example above demonstrates this: the original statement about multiples of 6 and 3 is true, but its converse and inverse are both false.

The Contrapositive Law

However, a conditional statement IS logically equivalent to its contrapositive:

p → q ≡ ∼q → ∼p

We can prove this using logical equivalence laws: ∼q → ∼p ≡ ∼(∼q) ∨ ∼p (Implication Law) ≡ q ∨ ∼p (Double Negative Law) ≡ ∼p ∨ q (Commutative Law) ≡ p → q (Implication Law)

This is extremely useful in mathematics! Sometimes it's easier to prove the contrapositive of a statement than the original statement directly.

Converse and Inverse Connection

Interestingly, the converse and the inverse of a conditional ARE logically equivalent to each other:

q → p ≡ ∼p → ∼q

This makes sense because the inverse (∼p → ∼q) is actually the contrapositive of the converse (q → p). Since a statement is equivalent to its contrapositive, the converse must be equivalent to the inverse.

💭 Challenge: Prove q → p ≡ ∼p → ∼q using the logical equivalence laws.

Here's a handy summary table:

Statement	Form	Logically Equivalent to Original (p → q)?
Original	p → q	Yes ✓
Converse	q → p	No ✗ (But equivalent to Inverse)
Inverse	∼p → ∼q	No ✗ (But equivalent to Converse)
Contrapositive	∼q → ∼p	Yes ✓

💭 Question: The statement "If I study chemistry, then I do not study IT" is logically equivalent to which of these? (i) It is necessary that I do not study IT for me to study chemistry. (ii) It is sufficient that I do not study IT for me to study chemistry. (Similar to Tutorial Test 1D, Q1d)

Answer: (i). The original is p → ∼q. Option (i) is also p → ∼q. Option (ii) is ∼q → p.

Biconditional: The Two-Way Street of Logic ⇔

In addition to the one-way implication (→), we have a two-way logical connector (Unit 1.2, Slide 8):

Symbol: ↔ (double arrow)

Meaning: p ↔ q means "p if and only if q" (often written "p iff q").

The biconditional p ↔ q is true only when p and q have the same truth value (both true or both false):

p	q	p ↔ q
T	T	T
T	F	F
F	T	F
F	F	T

The Biconditional Law

There's an important relationship between the biconditional and the conditional (Unit 1.2, Slide 9):

p ↔ q ≡ (p → q) ∧ (q → p)

In words: "p if and only if q" means "p implies q AND q implies p."

We can verify this with a truth table (Unit 1.2, Slide 10):

p	q	p ↔ q	p → q	q → p	(p → q) ∧ (q → p)
T	T	T	T	T	T
T	F	F	F	T	F
F	T	F	T	F	F
F	F	T	T	T	T

As you can see, the columns for p ↔ q and (p → q) ∧ (q → p) are identical.

Natural Language Expressions for Biconditionals

Just like with conditionals, biconditionals have several equivalent expressions in natural language (Unit 1.2, Slide 11):

p if and only if q
p iff q
p is necessary and sufficient for q
p implies q and q implies p
p is equivalent to q (in the context of logic)

💭 Question: We previously did the truth table for p ↔ (∼p ↔ ∼q). Based on the table, is this statement form a tautology, contradiction, or neither? (Similar to Tutorial Test 1A, Q2)

Answer: It's a tautology because the final column contains all T's.

💭 Another question: Is (p ∧ q) ↔ q logically equivalent to p ∨ ∼q? Justify with a truth table. (Similar to Tutorial Test 1F, Q2)

Arrow Abbreviations in Mathematical Writing 📝

In less formal mathematical writing, you might encounter these alternative notations (Unit 1.2, Slide 12):

p ⇒ q means p implies q (p → q)
p ⇐ q means p is implied by q (q → p)
p ⇔ q means p is equivalent to q (p ↔ q)

Updated Order of Operations ⚙️

With our new connectives, we can update our order of operations (Unit 1.2, Slide 13):

∼ (Highest precedence)
∧, ∨
→, ↔ (Lowest precedence

Notes, Divided By Lessons (0/3)

Part 1: Foundations - The Language of Mathematics: Logic (0/2)

Part 2 (0/1)

Topic 1.2: Conditional Statements - The "If-Then" World of Logic