Logical connectives, simple and compound statements

So far we have looked at single (simple) statements. In practice we very often build bigger statements out of smaller ones by joining them with connecting words like "and," "or," "if ... then," "if and only if," and "not." These joining words are called logical connectives.

Simple vs. compound statements. A statement that cannot be broken down any further into smaller statements is a simple statement. A statement built by joining two or more simple statements with a connective is a compound statement. For instance, "3 is a prime and 4 is an even number" is compound (it splices together "3 is a prime" and "4 is an even number" with and), whereas "3 and 5 are twin primes" is simple — even though it mentions two numbers, it expresses one indivisible fact about the pair, not a join of two separate statements.

We now go through each of the five connectives one at a time, and for each one independently work out its truth table from its meaning.

1. Conjunction ($p \wedge q$, read "$p$ and $q$"). Joining $p$ and $q$ with "and" gives their conjunction. The combined statement is true precisely when both $p$ and $q$ are true on their own — if even one of them fails, the whole "and" statement fails.

2. Disjunction ($p \vee q$, read "$p$ or $q$"). Joining $p$ and $q$ with "or" gives their disjunction. This is true whenever at least one of $p, q$ is true, and it fails only in the single case where both are false.

3. Conditional / implication ($p \rightarrow q$, read "if $p$ then $q$," also written $p \Rightarrow q$). Here $p$ is called the hypothesis (or antecedent) and $q$ the conclusion (or consequence). The only situation where "if $p$ then $q$" should be judged false is when $p$ actually holds but $q$ does not — a true cause failing to produce its promised effect. In every other combination the conditional is treated as true (including, perhaps surprisingly, when $p$ itself is false — a false hypothesis doesn't break the promise).

The same relationship $p \rightarrow q$ can be phrased in several equivalent ways: "$p$ is sufficient for $q$," "$q$ is necessary for $p$," "$p$ implies $q$," "$q$ follows from $p$," or "$p$ only if $q$."

4. Biconditional / double implication ($p \leftrightarrow q$, read "$p$ if and only if $q$," also written $p \Leftrightarrow q$). This combines "$p \rightarrow q$" with its reverse, so it is true exactly when $p$ and $q$ agree — that is, when they carry the same truth value (both true or both false) — and false whenever they disagree.

5. Negation ($\sim p$, read "not $p$"). Unlike the other four, negation acts on a single statement: it simply flips the truth value of $p$.

Applying negation twice returns the original statement: $\sim(\sim p) = p$.

Worked Example 1 — writing compound statements symbolically. Express each sentence symbolically, without worrying about whether it is actually true.

1. "2 is an even number and 25 is a perfect square." Let $p$: 2 is an even number; $q$: 25 is a perfect square. Since the two parts are joined by "and," the symbolic form is $p \wedge q$.
2. "A school is open or there is a holiday." Let $p$: the school is open; $q$: there is a holiday. Joined by "or," the symbolic form is $p \vee q$.
3. "Delhi is in India but Dhaka is not in Sri Lanka." Here "but" functions the same way as "and." Let $p$: Delhi is in India; $q$: Dhaka is in Sri Lanka. Since the second part is negated ("is not in Sri Lanka"), the symbolic form is $p \wedge \sim q$.
4. "$3+8 \ge 12$ if and only if $5\times4 \le 25$." Let $p$: $3+8\ge12$; $q$: $5\times4\le25$. The connective "if and only if" gives the symbolic form $p \leftrightarrow q$. Worked Example 2 — finding the truth value of a compound statement. For each sentence, assign $p$ and $q$, find their individual truth values, then evaluate the connective.

(i) "3 is a prime number and 4 is a rational number." $p$: 3 is prime — true. $q$: 4 is rational — true. Symbolic form $p\wedge q$, so the truth value is $T \wedge T = T$.

(ii) "All flowers are red or all cows are black." $p$: all flowers are red — false (plenty of flowers aren't red). $q$: all cows are black — false (plenty of cows aren't black). Symbolic form $p \vee q$, so the truth value is $F \vee F = F$.

(iii) "If Mumbai is in Maharashtra then Delhi is the capital of India." $p$: Mumbai is in Maharashtra — true. $q$: Delhi is the capital of India — true. Symbolic form $p\rightarrow q$, so the truth value is $T\rightarrow T = T$.

(iv) "Milk is white if and only if the Sun rises in the West." $p$: milk is white — true. $q$: the Sun rises in the West — false (it rises in the East). Symbolic form $p\leftrightarrow q$, so the truth value is $T\leftrightarrow F = F$.

Worked Example 3 — evaluating statement patterns. Suppose $p, q$ are true and $r, s$ are false. Evaluate each pattern by substituting the given truth values and simplifying step by step. …