Lecture 3 - Propositional logic

Connectives

• $\neg$ - negation (not)
• $\wedge$ - conjunction (and)
• $\vee$ - disjunction (or)
• $\to$ - implication
• $\leftrightarrow$ - equivalence The negation is a unary connective and all the others are binary connectives.

Decreasing order of precedence

$\neg \wedge \vee \to \leftrightarrow$

New connectives

• $\uparrow$ - nand - $\neg (p\wedge q)$
• $\downarrow$ - nor - $\neg (p\vee q)$
• $\oplus$ - xor (exclusive or) $\neg (p\leftrightarrow q)$

Logical equivalence

The formulas U and V are logically equivalent, if they have identical truth tables: $U\equiv V$

Exercise

$X=(p\wedge q\to r)\to (p\to r)\wedge q\equiv (\neg (p\wedge q)\vee r)\to (\neg p\vee r)\wedge q\equiv$ $\equiv \neg (\neg (p\wedge q)\vee r)\wedge (\neg p\vee r)\wedge q\equiv (p\wedge q\wedge \neg r)\vee (\neg p\vee r)\wedge q\equiv$ $(p\wedge q\wedge \neg r)\vee (\neg p\wedge q)\vee (r\wedge q)$

Homework