Logical Form and Logical Equivalence

MATH1061

Introduction to Logical Form

1. ~ p $\wedge$ q

2. ~ q $\vee$ ~ p $\wedge$ r

3. It is not snowing but it is cold

4. It could be cold and snowing or it could be sunny.

Logical Form Notation

Statement Variables

Logical connectives are ways to connect one number from another.

AND/BUT = $\wedge$

OR = $\vee$

NOT = $\sim$

Inclusive and Exclusive OR

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The symbol $\vee$ means “inclusive or:. So $p\vee q$ is true when $p$ is true, or $q$ is true or both are true. Sometimes when the word “or” is used, the meaning is “one or the other, but not both”.

This is “exclusive or”. Exclusive or is often written using the logical connective $\oplus$. | P | Q | R | Statement Form $(p\wedge \sim q)\vee (q\wedge r)$| | --- | --- | --- | --- | -------------- | | t | t | t | f | t | | t | t | f | f | f | | t | f | t | t | f | | t | f | f | t | f | | f | t | t | f | t | | f | t | f | f | f | | f | f | t | t | f | | f | f | f | t | f |

De Morgan’s Laws

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De Morgan's Laws

Important logical equivalences for the negations of statements involving “and” and “or”. $\sim (p\wedge q)\equiv \sim p\vee \sim q$

$\sim (p\vee q)\equiv \sim p\wedge \sim q$

Use the laws of logical equivalence to show that:

$\sim ((\sim p\wedge q)\vee (\sim p\wedge \sim q))\equiv p$

$$\equiv \sim (\sim p\wedge q)\wedge \sim (\sim p\wedge \sim q)\text{De Morgan’s Law}$$

$\equiv (\sim (\sim p)\vee \sim q)\wedge (\sim (\sim p)\vee \sim (\sim q))$