Logic is a branch of mathematics that studies reasoning and arguments. It uses formal languages and systems of inference to represent mathematical statements, propositions, and rules of reasoning. A statement is a declarative sentence that can be either true or false. A simple statement conveys a single idea, while a compound statement conveys two or more ideas by connecting simple statements with connectives like "and", "or", and "if...then". The truth value of compound statements depends on the truth values of its simple statements and connectives, which can be represented using truth tables.

2. Logic is a branch of mathematics that studies the principles of
reasoning and inference. It is concerned with the study of valid
reasoning and argumentation, and the methods of evaluating the
correctness of arguments.
What is Logic?
In mathematical logic, we use formal languages and systems of
inference to study the structure and properties of mathematical
reasoning. We use symbols and logical operators to represent
mathematical statements and propositions, and we use rules of
inference to derive new statements from existing ones.
A
B
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D

3. Every language contains different types of sentences, such as
statements, questions, and commands.
Logic Statements
1. Is the test today?
2. Go get the newspaper.
3. This is a nice car.
4. Denver is the capital of Colorado.
A
B
C
D
Statement is a declarative sentence that is either true or false, but not
both true and false.

4. Determine whether each sentence is a statement.
Logic Statements
1. Florida is a state in the United States.
2. How are you?
3. 99 + 2 is a prime number.
4. x + 1 = 5.
A
B
C
D

5. Simple and Compound Statements
Connecting simple statements with words and phrases such as and,
or, if…. then, and if and only if creates a compound statement.
A
B
C
D
A simple statement is a statement that conveys a single idea.
A compound statement is a statement that conveys two or more
ideas.
George Boole used symbols such as p, q, r, and s to represent
simple statements and the symbols ∧, ∨, ∼, →, ↔ to represent
connectives.

6. Simple and Compound Statements
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B
C
D
Statement Connective Symbolic Form Type of Statement
not p not ∼ 𝒑 negation
p and q and 𝒑 ∧ 𝒒 conjunction
p or q or 𝒑 ∨ 𝒒 disjunction
If p, then q If… then 𝒑 → 𝒒 conditional
p if and only if q if and only if 𝒑 ↔ 𝒒 biconditional

7. Simple and Compound Statements
The truth value of a compound statement depends on the truth
values of its simple statements and its connectives.
A
B
C
D
The truth value of a simple statement is either true (T) or false (F)
A truth table is a table that shows the truth value of a compound
statement for all possible truth values of its simple statements.

8. Simple and Compound Statements
Given: p : Today is Friday
q : It is raining
r : I am going to a movie
s : I am not going to the basketball game
1. ∼ 𝒑 6. 𝒑 →∼ 𝒓
2. 𝒑 ∧ 𝒒 7.∼ 𝒒 → 𝒓
3. 𝒒 ⟶ 𝒔 8. 𝐬 ⟶∼ 𝒑
4. ∼ 𝒒 ∧ 𝒓 9. ∼ 𝒓
5. 𝒓 ∨∼ 𝒔 10. ∼ 𝒔 ∨∼ 𝒓
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B
C
D
Rewrite the following statements:

9. Compound Statements and Groupings
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B
C
D
If a compound statement is written in symbolic form, then the
parentheses are used to indicate which simple statements are
grouped together.
Symbolic Form The parentheses indicate that
𝒑 ∧ (𝒒 ∨∼ 𝒓) q and ∼ 𝒓 are grouped together
(𝒑 ∧ 𝒒) ∨ 𝒓 p and q are grouped together
(𝒑 ∧∼ 𝒒) → (𝒓 ∨ 𝒔)
p and ∼ 𝒒 are grouped together
r and s are also grouped together

10. Compound Statements and Groupings
A
B
C
D
Let’s translate Compound Statements
Let p, q and r represent the following:
p : You get a promotion
q : You complete the training
r : You will receive a bonus
1. (𝒑 ∧ 𝒒) → 𝒓 2. ∼ 𝒒 → (∼ 𝒑 ∧∼ 𝒓)

11. Compound Statements and Groupings
A
B
C
D
The truth table of Conjunction
p q 𝒑 ∧ 𝒒
T T T
T F F
F T F
F F F
The conjunction 𝒑 ∧ 𝒒 is true if and only if both p and q are true.

12. Compound Statements and Groupings
A
B
C
D
The truth table of Disjunction
p q 𝒑 ∨ 𝒒
T T T
T F T
F T T
F F F
The disjunction 𝒑 ∨ 𝒒 is true if and only if p is true, q is true or both p
and q are true.

13. Compound Statements and Groupings
A
B
C
D
Determine the truth values of the following statement.
1. 7 ≥ 5.
2. 5 is a whole number and 5 is an even number.
3. 2 is a prime number and 2 is an even number.
4. 9 is 3 or the factor of 9 is 3.
5. A square is a rectangle and rectangle is a square.

14. Thank you ☺