2.1 Logical Form and Equivalence
Statement (Proposition):
Ex. Are the following statements? Why or why not?
2+2=4
2+2=5
He goes to college.
x+y<0
Compound statements:
The three common logical connectors are:
Connector Name
Connector symbol
We can also use parentheses as in algebra to indicate the order in which connectors should be used.
Ex. Translate the following into symbolic logic statements. Let h = “It is hot” and s = “It is sunny”
a. It is hot and it is sunny.
b. It is not hot but it is sunny.
c. It is neither hot nor sunny.
Ex. Translate the following into symbolic logic statements. Let 𝑝 represent 0 < 𝑥, 𝑞 be 𝑥 < 3, and 𝑟 be 𝑥 = 3.
a. 𝑥 ≤ 3
b. 0 < 𝑥 < 3
c. 0 < 𝑥 ≤ 3
Truth Tables: Allow us to explore all possible truth values (T or F) of a compound statement
Basic Truth Tables for the three logical connectors. When constructing a truth table, we start with columns for the
individual logical variables in the statement. We need to have enough rows in the truth table so that we can explore
all possible combinations of true and false individual variables. We will also establish a basic pattern for tables that
are easier to read.
Negation (~𝒑)
2.1 Logical Form and Equivalence
Conjunction (𝒑 ∧ 𝒒):
Disjunction (𝒑 ∨ 𝒒):
When creating a truth table for a more complex statement, we should still start with columns for the individual
________________________. The final column in our truth table should be the ______________________________________. To get
there, we need to create a column each time a new _____________________________________ is added to the statement.
Ex. Create a truth table for the following compound statement: (𝑝 ∧ 𝑞) ∨ ~𝑟
2.1 Logical Form and Equivalence
Logical Equivalence:
To test for logical equivalence:
Ex. Test if ~(𝑝 ∧ 𝑞) and ~𝑝 ∧ ~𝑞 are logically equivalent.
DeMorgan’s Laws: Give rules for the negation of a conjunction and disjunction
Ex. Use DeMorgan’s Laws to negate the following statements:
a. John is 6 feet tall and weighs at least 200 pounds.
b. The bus was late and Tom’s watch was not slow.
Tautology:
Contradiction:
Ex. Construct truth tables to determine if 𝑝 ∨ ~𝑝 and 𝑝 ∧ ~𝑝 are tautologies or contradictions.
2.1 Logical Form and Equivalence
Simplifying Statements: We can use common logical equivalences to simplify complex statements
Given any statement variables p, q, and r, a tautology t and a contradiction c, the following logical equivalences
hold.
1. Commutative laws
𝑝∧𝑞 ≡𝑞∧𝑝
𝑝∨𝑞 ≡𝑞∨𝑝
2. Associative laws
(𝑝 ∧ 𝑞) ∧ 𝑟 ≡ 𝑝 ∧ (𝑞 ∧ 𝑟)
(𝑝 ∨ 𝑞) ∨ 𝑟 ≡ 𝑝 ∨ (𝑞 ∨ 𝑟)
3. Distributive laws
𝑝 ∧ (𝑞 ∨ 𝑟) ≡ (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟)
𝑝 ∨ (𝑞 ∧ 𝑟) ≡ (𝑝 ∨ 𝑞) ∧ (𝑝 ∧ 𝑟)
4. Identity laws
𝑝∧t≡𝑝
𝑝∨ c≡𝑝
5. Negation laws
𝑝 ∨∼ 𝑝 ≡ t
𝑝 ∧∼ 𝑝 ≡ c
6. Double negative law
7. Idempotent laws
8. Universal bound laws
9. De Morgan’s Laws
10. Absorption laws
11. Negations of t and c
∼ (∼ 𝑝) ≡ 𝑝
𝑝∧𝑝≡𝑝
𝑝∨𝑝≡𝑝
𝑝∨t≡t
𝑝∧c≡c
∼ (𝑝 ∧ 𝑞) ≡∼ 𝑝 ∨∼ 𝑞
∼ (𝑝 ∨ 𝑞) ≡∼ 𝑝 ∧∼ 𝑞
𝑝 ∨ (𝑝 ∧ 𝑞) ≡ 𝑝
𝑝 ∧ (𝑝 ∨ 𝑞) ≡ 𝑝
∼t≡c
∼c≡t
Ex. Simplify the following complex statement to achieve the desired simplication. Clearly indicate which law you
are using for each step:
~(~𝑝 ∧ 𝑞) ∧ (𝑝 ∨ 𝑞) ≡ 𝑝