Conditional and Bi-conditional Statements

Conditional and Bi-conditional Statements

1. Conditional Statements (Implication)

A conditional statement is of the form:

Read as: “If $p$, then $q$”

• $p$: Hypothesis (antecedent)
• $q$: Conclusion (consequent)

This statement is false only when the hypothesis is true and the conclusion is false. In all other cases, it is true.

Truth Table for $p \rightarrow q$:

Examples:

1. "If it rains, then the streets get wet."

   • Let $p$: It rains
   • Let $q$: Streets get wet
   • Form: $p \rightarrow q$

2. "If a number is divisible by 4, then it is even."

   • $p$: Number is divisible by 4
   • $q$: Number is even
   • $p \rightarrow q$: True

2. Variations of Conditional Statements

• Converse: $q \rightarrow p$
• Inverse: $\neg p \rightarrow \neg q$
• Contrapositive: $\neg q \rightarrow \neg p$

Important:

• $p \rightarrow q$ is logically equivalent to its contrapositive $\neg q \rightarrow \neg p$
• The converse and inverse are logically equivalent to each other, but not necessarily to the original statement

3. Bi-conditional Statements (Double Implication)

A bi-conditional statement is of the form:

Read as: “$p$ if and only if $q$”
This means both:

• $p \rightarrow q$
• $q \rightarrow p$

The statement is true only when both $p$ and $q$ have the same truth value.

Truth Table for $p \leftrightarrow q$:

Examples:

1. "You can enter the contest if and only if you are 18 or older."

   • $p$: You can enter the contest
   • $q$: You are 18 or older
   • $p \leftrightarrow q$: True when both statements agree in truth

2. "A number is even if and only if it is divisible by 2."

   • $p$: Number is even
   • $q$: Number is divisible by 2
   • Logical equivalence

4. Logical Equivalences Involving Conditionals

• $p \rightarrow q \equiv \neg p \lor q$
• $p \leftrightarrow q \equiv (p \rightarrow q) \land (q \rightarrow p)$

These equivalences are useful in simplifying logical expressions and in constructing truth tables.