Thinking Mathematically

Logic3.2 Compound Statements and Connectives

“Compound” Statements

Simple statements can be connected with “and”, “Either … or”, “If … then”, or “if and only if.” These more complicated statements are called “compound.”

Symbolic Logic

Name Symbolic Form

Read

Negation ~p not p

Conjunction p/\q p and q

Disjunction p\/q p or q

Symbolic Logic (cont.)

Name Symbolic Form Read

Conditional p q • if p then q•p is sufficient for q•q is necessary for p

Biconditional p q • p if and only if q•p is necessary and sufficient for q•q is necessary and sufficient for p

Examples

Exercise Set 3.2 #3, #7p: I’m leaving.q: You’re staying.You’re staying and I’m not leaving.

p: I studyq: I pass the courseI study or I pass the course.

Examples

Exercise Set 3.2 #11, #23p: This is an alligator.q: This is a reptile.If this is an alligator, then this is a reptile.

p: You are human.q: You have feathers.Being human is sufficient for not having feathers.

Examples

Exercise Set 3.2 #35, #49p: The heater is workingq: The house is coldp \/ ~q

p: Romeo loves Juliet.q: Juliet loves Romeo.~(p /\ q)

Dominance of Connectives

1. Negation (~)

2. Conjunction/Disjunction (/\, \/)3. Conditional ()4. Biconditional ()

• The most dominant is applied last• Analogous to order of operations in algebra

Examples

Exercise Set 3.2 #59, #79p: The temperature outside is freezing.q: The heater is working.r: The house is cold.The temperature outside is freezing and the heater is

working, or the house is cold.

p: The temperature is above 85o

q: We finished studyingr: We go to the beach.~r ~(p /\ q)

Examples

• Exercise Set 3.2 #85I miss class if and only if it’s not true that both I

like the teacher and the course is interesting.

Thinking Mathematically

Logic3.2 Compound Statements and Connectives