Implications and Quantifiers

1. The Conditional Statement (Implications)

A compound statement of the form "if p, then q" is called a conditional statement or an implication. It is denoted by $p \implies q$.

• $p$ is called the hypothesis (or antecedent).
• $q$ is called the conclusion (or consequent).

An implication $p \implies q$ is false in only one case: when the hypothesis $p$ is true and the conclusion $q$ is false. In all other cases, the implication is true.

• Example: "If you score above 90%, then you will get an A." This promise is only broken (the statement is false) if you score above 90% (hypothesis is true) but you do not get an A (conclusion is false).

Truth Table for Implication:

$p$	$q$	$p \implies q$
T	T	T
T	F	F
F	T	T
F	F	T

2. The Biconditional Statement (If and Only If)

A compound statement of the form "p if and only if q" (often abbreviated as "p iff q") is called a biconditional statement. It is denoted by $p \iff q$. This statement is the conjunction of two implications: $(p \implies q) \land (q \implies p)$.

A biconditional statement $p \iff q$ is true only when $p$ and $q$ have the same truth value (both true or both false).

• Example: "A triangle is equilateral if and only if all its three angles are equal."

Truth Table for Biconditional:

$p$	$q$	$p \iff q$
T	T	T
T	F	F
F	T	F
F	F	T

3. Quantifiers

Quantifiers are phrases that indicate the scope of a statement, i.e., how many objects a statement applies to.

A. The Universal Quantifier ("For All")

The symbol $\forall$ (an inverted 'A') stands for "for all" or "for every". It is called the universal quantifier. It makes a claim that a property is true for every object in a given set.

• Example: $\forall x \in N, x+1 > x$. (For every natural number x, x+1 is greater than x.) This is a true statement.

B. The Existential Quantifier ("There Exists")

The symbol $\exists$ (a reversed 'E') stands for "there exists" or "for some". It is called the existential quantifier. It makes a claim that there is at least one object in a given set for which a property is true.

• Example: $\exists x \in Z, x^2 = 4$. (There exists an integer x such that $x^2=4$.) This is a true statement (since $x=2$ or $x=-2$ works).

4. Negating Quantified Statements

Negating statements with quantifiers involves swapping the quantifier and negating the property.

• The negation of $\forall x, P(x)$ is $\exists x, \sim P(x)$.

  • Statement: "All dogs have tails." ($\forall$ dogs, it has a tail).
  • Negation: "There exists a dog that does not have a tail." ($\exists$ a dog, it does not have a tail).

• The negation of $\exists x, P(x)$ is $\forall x, \sim P(x)$.

  • Statement: "There is a student who scored 100%." ($\exists$ a student, they scored 100%).
  • Negation: "All students did not score 100%." or "No student scored 100%." ($\forall$ students, they did not score 100%).