Contemporary Mathematics Topic 6 Statements Logic Beginning with

Contemporary Mathematics Topic 6: Statements

Logic › Beginning with this topic, we begin a detailed look at formal logic, and the mathematics of logic. › In this section, we will study the basic building block of logic: statements. › A statement is a sentence than can be classified as either true or false. Note that a statement cannot be both true and false simultaneously. Also, statements need to be stated declaratively.

Example 1: › Which of the following are statements? › A) The sky is blue. › B) What day is it? › C) I am either silly or stupid. › D) Wow! › E) This statement is false. › (The last one is known as a paradox. It cannot be classified as true or false. )

Compound Statements ›

Negations ›

Example 2: ›

De Morgan’s Laws ›

Quantifiers › Some statements are harder to negate. These statements have quantifiers in them. › Words such as “all”, “none”, “every”, “for all”, etc. are called universal quantifiers. › Words such as “some”, “there exists”, “for at least one”, etc. are called existential quantifiers. › Positive universal quantifiers, when negated, become negative existential quantifiers, and vice-versa. Likewise, negative universal quantifiers become positive existential quantifiers when negated.

Illustrations of Quantifier Negations › For instance, “All of us like peanuts” when negated becomes “Some of us do not like peanuts”/”There is at least one of us (or there exists someone) that does not like peanuts. ” › Or “Some sunsets are worth watching” when negated becomes “No sunsets are worth watching”. › All Some are not › None Some are › Remember this; it will save you many mistakes!

Example 3: › Write the negation of each of the following statements, simplifying as much as possible. › A) “All freshmen are required to have a meal plan. ” › B) “In our group, there exists a couple of people who do not like meat. ” › C) “Either all of us are going to eat out tonight, or I will buy none of you birthday gifts. ”