Truth Tables and Logical Statements

Truth Tables

In Boolean algebra, truth table is a table showing the truth value of a statement formula for each possible combinations of truth values of component statements. A statement is a declarative sentence which has one and only one of the two possible values called truth values. Truth values are true and false denoted by the symbols T and F respectively, sometimes also denoted by symbols 1 and 0. Since we allow only two possible truth values, this logic is called two-valued logic.

Truth Tables for Unary Operations

Truth table is a powerful concept that constructs truth tables for its component statements. Whereas, the unary logical operations are those operations which contain only one logical variable. Let’s study in detail.

Truth Table for Logical True

Logical true returns a true value for whatever every input. Its truth table is:

P T(P)
T T
F T

Truth Table for Logical False

Logical false gives a false value for whatever the input is. Its truth table is as follows.

P T(P)
T F
F F

Truth Table for Negation

Logical negation is a unary operation which typically returns the opposite value of a proposition. If input is false, then output is true and vice versa. It is represented by NOT, or ~p. The truth table for NOT is given below.

P ∼P
T F
F T

Truth Tables for Binary Operations

In logical mathematics, binary operations are the logical operations that have two logical input variables. The truth tables of the most important binary operations are given below.

Truth Table for Conjunction

A conjunction is a binary logical operation which results in a true value if both the input variables are true. This operator is represented by P AND Q or P ∧ Q or P . Q or P & Q, where P and Q are input variables. Its truth table is given below:

P Q P ∧ Q
T T T
T F F
F T F
F F F

Truth Table for Disjunction

Logical disjunction returns a true when at least input operands are true, i.e. either one of them or both are true.

It is denoted by the symbols P OR Q, P ∨ Q or P + Q. Its truth table is shown below:

P Q P ∨ Q
T T T
T F T
F T T
F F F

Truth Table for Implication

Logical implication typically produces a value of false in singular case that the first input is true and the second is either false or true. It is associated with the condition, “if P then Q” [Conditional Statement] and is denoted by P → Q or P ⇒ Q. The truth table for implication is as follows:

P Q P → Q
T T T
T F F
F T T
F F T

BiConditional Truth Table

The equivalence P \leftrightarrow Q is true if both P and Q are true OR both P and Q are false. It is associated with the condition, “P if and only if Q” [BiConditional Statement] and is denoted by P \leftrightarrow Q. The truth table is as follows:

P Q P \leftrightarrow Q
T T T
T F F
F T F
F F T

Logic and Truth Tables

Here we will discuss the logic tables operations with truth tables. The propositional logic truth tables are the standard one. So we can’t change the propositional value.

Logical NAND

The NAND is a binary logical operation which is similar to applying NOT on AND operation. In other words, NAND produces a true value if at least one of the input variables is false. It is denoted by P NAND Q or P | Q or P ↑ Q. Have a look at its truth table.

P Q (P ∧ Q) And
T T F
T F T
F T T
F F T

Logical NOR

The logical NOR is a logical operation which is obtained by applying a NOT operation to an OR operation. We can say that NOR results in a true value if both the input variables are false. It is represented by P NOR Q or P ↓ Q. Take a look at its truth table.

P Q (P ∨ Q) OR
T T F
T F F
F T F
F F T

Steps to Solve Logical Expressions

To solve logical expressions we proceed as follows

1. First, solve the parentheses if any.

2. Next, solve the NOT operator.

3. Then the AND operator.

4. And lastly the OR operator.

Logical Thinking and Statements

Logical thinking is the ability to understand the facts and concepts of the quantity, ideas, number system and other relationship between the ideas and the concepts. It is an ability to view a concept in a different method. It is a good skill in math solving. It deals with the ability to solve mathematical problems, planning, predicting outcomes, voluntary risk etc.

The standards of logical-mathematical intelligence concepts

The ability for Problem Solving

Skills of Communication

Ability of Reasoning

Viewing Geometry and Spatial Sense

Methods of Measurement

Statistics and Probability

Patterns and Relationships, and many more

Important Laws followed by Statements

If A, B and C are statements, then

Commutative Laws A ∨ B = B ∨ A and A ∧ B = A ∧ B
Associative Laws A ∨ (B ∨ C) = (A ∨ B) ∨ C and

A ∧ (B ∧ C) = (A ∧ B) ∧ C

Distributive Laws A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C) and

A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)

Idempotent Laws A ∨ B = A and

A ∧ B = A

Identity Laws A ∨ t = t,

A ∧ t = A,

A ∨ m = A and

A ∧ m = m

Complement Laws A ∨ (~A) = t,

A ∧ (~A) = m,

~t = m and

~m = t

De-Morgan’s Law ~(A ∨ B) = (~A) ∧ (~B) and

~(A ∧ B) = (~A) ∨ (~B)

Truth Tables Examples

Example 1: Find the logical truth table for given values using conjunction.

If P: F F T F T and Q: F T T T F

Solution:

P Q P ∧ Q
F F F
F T F
T T T
F T F
T F F

Example 2: Construct the truth table for ~P∨∼Q and ∼(P∧Q).

Solution:

P Q ~P ~Q ~P∨∼Q (P∧Q) ~(P∧Q)
T T F F F T F
T F F T T F T
F T T F T F T
F F T T T F T

Example 3: Which of the following is the contrapositive of ‘if two triangles are identical, then these are similar’?

A) If two triangles are not similar, then they are not identical.

B) If two triangles are not identical, then these are not similar.

C) If two triangles are not identical, then these are similar.

D) If two triangles are not similar, then these are identical.

Solution:

Consider the following statements

p: Two triangles are identical.

q: Two triangles are similar.

Clearly, the given statement in symbolic form is ⇒ q.

Therefore, its contrapositive is given by ⇒ p

Now,

∼p: Two triangles are not identical.
q: Two triangles are not similar.
 ~q    ~p: If two triangles are not similar, then these are not identical.

Example 4: The statement ~(p↔ ~q) is

A) equivalent to p↔q

B) equivalent to ~p↔q

C) a tautology

D) a fallacy

Solution: 

p q ∼q p↔∼q ∼(p↔∼q) p↔q
T T F F T T
T F T T F F
F T F T F F
F F T F T T

Example 5: Let S be a non-empty subset of R. Consider the following statement: P: There is a rational number x ∈ S such that x>0. Which of the following statements is the negation of the statement p?

A) x ∈ S and x ≤ 0 ⇒ x is not rational.

B) There is a rational number x ∈ S such that x ≤ 0.

C) There is no rational number x ∈ S such that x ≤ 0.

D) Every rational number x ∈ S satisfies x ≤ 0.

Solution: 

P: There is a rational number x ∈ S such that x > 0.

~ P: Every rational number x ∈ S satisfies x ≤ 0.

Also read

Linear programming problems

Arithmetic mean