Definitions [6]
A statement which cannot be split further into two or more statements is called a simple statement. If a statement is the combination of two or more simple statements, then it is called a compound statement.
Two statement patterns are said to be equivalent if their truth tables are identical. If statement patterns A and B are equivalent, we write it as A ≡ B.
A statement is a declarative (assertive) sentence which is either true or false, but not both simultaneously. Statements are denoted by p, q, r, ....
Open Statement:
An open sentence is a sentence whose truth can vary depending on conditions not stated in the sentence.
Each statement is either true or false. If a statement is true, then its truth value is 'T', and if the statement is false, then its truth value is F.
The words or phrases which are used to connect two statements are called logical connectives.
Eg. 'and', 'or', 'if ..... then', 'if and only if ', 'not''.
Two compound statements s1 and s2 are said to be duals of each other if one can be obtained from the other by:
-
Replacing ∧ (and) by ∨ (or)
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Replacing ∨ (or) by ∧ (and)
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Replacing T (tautology) by F (contradiction)
-
Replacing F (contradiction) by T (tautology)
while keeping negations unchanged.
Theorems and Laws [1]
Using truth table prove that ~ p ˄ q ≡ ( p ˅ q) ˄ ~ p
~ p ∧ q ≡ (p ∨ q) ∧ ~ p
| 1 | 2 | 3 | 4 | 5 | 6 |
| p | q | ~p | ~p ∧ q | (p ∨ q) | (p ∨ q) ∧ ~p |
| T | T | F | F | T | F |
| T | F | F | F | T | F |
| F | T | T | T | T | T |
| F | F | T | F | F | F |
In the above truth table, the entries in the columns 4 and 6 are identical.
∴ ~p ∧ q ≡ (p ∨ q) ∧ ~p.
Key Points
| Type | Meaning |
|---|---|
| Tautology | Always True |
| Contradiction | Always False |
| Contingency | Sometimes T, Sometimes F |
| Symbol | Meaning |
|---|---|
| ∀ | For all (Universal quantifier) |
| ∃ | There exists (Existential quantifier) |
Quantified statement: An open sentence with a quantifier becomes a statement and is called a quantified statement.
| Circuit Type | Logical Form |
|---|---|
| Series | p ∧ q |
| Parallel | p ∨ q |
Switch ON = 1
Switch OFF = 0
For p → q:
| Type | Form |
|---|---|
| Converse | q → p |
| Inverse | ∼p → ∼q |
| Contrapositive | ∼q → ∼p |
| Type | Given Statement | Negation | Symbolic Form |
|---|---|---|---|
| Negation of Conjunction | p ∧ q | Not p or Not q | ~(p ∧ q) ≡ ~p ∨ ~q |
| Negation of Disjunction | p ∨ q | Not p and Not q | ~(p ∨ q) ≡ ~p ∧ ~q |
| Negation of Implication | p → q | p and Not q | ~(p → q) ≡ p ∧ ~q |
| Negation of Biconditional | p ↔ q | (p and Not q) or (q and Not p) | ~(p ↔ q) ≡ (p ∧ ~q) ∨ (q ∧ ~p) |
| Negation of Quantified Statement | ∀ x P(x) / ∃ x P(x) | Replace “all” by “some” and vice versa, and negate P(x) | ~(∀ x P(x)) ≡ ∃x ~P(x) ~(∃x P(x)) ≡ ∀x ~P(x) |
| Sr. No. | Law Name | Logical Form |
|---|---|---|
| 1 | Idempotent Law | p ∧ p ≡ p p ∨ p ≡ p |
| 2 | Commutative Law | p ∧ q ≡ q ∧ p p ∨ q ≡ q ∨ p |
| 3 | Associative Law | p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r ≡ p ∧ q ∧ r p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r ≡ p ∨ q ∨ r |
| 4 | Distributive Law | p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) |
| 5 | De Morgan’s Laws | ~(p ∧ q) ≡ ~p ∨ ~q ~(p ∨ q) ≡ ~p ∧ ~q |
| 6 | Identity Laws | p ∧ T ≡ p p ∨ F ≡ p p ∧ F ≡ F p ∨ T ≡ T |
| 7 | Complement Laws | p ∧ ~p ≡ F p ∨ ~p ≡ T |
| 8 | Absorption Laws | p ∨ (p ∧ q) ≡ p p ∧ (p ∨ q) ≡ p |
| 9 |
Conditional Law |
p → q ≡ ~p ∨ q |
| 10 | Biconditional Law |
p ↔ q ≡ (p → q) ∧ (q → p) ≡ (~p ∨ q) ∧ (~q ∨ p) |
| Connective | Symbol | Name | True When |
|---|---|---|---|
| and | ∧ | Conjunction | Both true |
| or | ∨ | Disjunction | At least one true |
| if...then | → | Conditional | False only when T → F |
| iff | ↔ | Biconditional | Same truth values |
| not | ~ | Negation | Opposite value |
Note: ~ (~ p) = p
Important Questions [45]
- Write the Following Statement in Symbolic Form and Find Its Truth Value
- If p, q, r are the statements with truth values T, F, T, respectively then find the truth value of (r ∧ q) ↔ ∼ p
- Using truth table prove that ~ p ˄ q ≡ ( p ˅ q) ˄ ~ p
- If p ˄ q = F, p → q = F, then the truth value of p and q is ______.
- If A = {1, 2, 3, 4, 5} then which of the following is not true?
- Write truth values of the following statements √5 is an irrational number but 3 +√5 is a complex number.
- If p ∧ q is F, p → q is F then the truth values of p and q are ________.
- Write truth values of the following statements : ∃ n ∈ N such that n + 5 > 10.
- Using truth tables, examine whether the statement pattern (p ∧ q) ∨ (p ∧ r) is a tautology, contradiction or contingency.
- Using truth table prove that ∼p ˄ q ≡ (p ˅ q) ˄ ∼p
- Using truth table, prove the following logical equivalence : (p ∧ q) → r ≡ p → (q → r)
- Write the following compound statement symbolically. Nagpur is in Maharashtra and Chennai is in Tamil Nadu.
- Write down the following statements in symbolic form : (A) A triangle is equilateral if and only if it is equiangular. (B) Price increases and demand falls
- Using the truth table, prove the following logical equivalence : p ↔ q ≡ (p ∧ q) ∨ (~p ∧ ~q)
- Using truth table, prove that ~ p ∧ q ≡ (p ∨ q) ∧ ~ p
- Write the following compound statement symbolically. If ΔABC is right-angled at B, then m∠A + m∠C = 90°.
- Construct the truth table for the statement pattern: [(p → q) ∧ q] → p
- Using truth table prove that p ↔ q = (p ∧ q) ∨ (~p ∧ ~q).
- Discuss the statement pattern, using truth table : ~(~p ∧ ~q) v q
- Inverse of statement pattern (p ∨ q) → (p ∧ q) is ________ .
- Examine whether the following logical statement pattern is a tautology, contradiction, or contingency.[(p → q) ∧ q] → p
- Write the converse and contrapositive of the statement — “If two triangles are congruent, then their areas are equal.”
- Using Truth Table Examine Whether the Following Statement Pattern is Tautology, Contradiction Or Contingency
- Write the dual of the following statements: (l) (p ∨ q) ∧ T (2) Madhuri has curly hair and brown eyes .
- Write the Dual of the Following Statements: Madhuri Has Curly Hair and Brown Eyes.
- Using truth table verify that: (p ∧ q)∨ ∼ q ≡ p∨ ∼ q
- The converse of contrapositive of ∼p → q is ______.
- The dual of statement t ∨ (p ∨ q) is ______.
- The negation of p ^ (q → r) is ______.
- Write converse, inverse and contrapositive of the following statement. If x < y then x2 < y2 (x, y ∈ R)
- The negation of p ∧ (q → r) is
- If a = {2, 3, 4, 5, 6}, Then Which of the Following is Not True?
- Without using the truth table show that P ↔ q ≡ (p ∧ q) ∨ (~ p ∧ ~ q)
- Without using truth tabic show that ~(p v q)v(~p ∧ q) = ~p
- Without using truth table prove that (p ∧ q) ∨ (∼ p ∧ q) v (p∧ ∼ q) ≡ p ∨ q
- Simplify the given circuit by writing its logical expression. Also, write your conclusion.
- Construct the New Switching Circuit for the Following Circuit with Only One Switch by Simplifying the Given Circuit:
- Construct the Simplified Circuit for the Following Circuit
- Find the symbolic form of the given switching circuit. Construct its switching table and interpret your result. diagram
- Construct the Switching Circuit for the Statement (P ∧ Q) ∨ (~ P) ∨ (P ∧ ~ Q)
- Find the symbolic form of the following switching circuit, construct its switching table and interpret it.
- Construct the switching circuit of the following: (∼ p ∧ q) ∨ (p ∧ ∼ r)
- Give an alternative equivalent simple circuit for the following circuit:
- Simplify the following circuit so that the new circuit has minimum number of switches. Also, draw the simplified circuit.
- Construct the Switching Circuit for the Following Statement : P V (~ P ∧ Q) V (- Q ∧ R) V ~ P
Concepts [12]
- Concept of Statements
- Truth Value of Statement
- Logical Connective, Simple and Compound Statements
- Statement Patterns and Logical Equivalence
- Tautology, Contradiction, and Contingency
- Duality
- Quantifier and Quantified Statements in Logic
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits
- Overview of Mathematical Logic
