Definitions [10]
Definition: Statements and Truth Values
A statement is a declarative sentence which is either true or false, but not both simultaneously.
- Statements are denoted by lower-case letters p, q, r, etc.
- The truth value of a statement is denoted by ‘1’ or ‘T’ for True and ‘0’ or ‘F’ for False.
- Open sentences, imperative sentences, exclamatory sentences and interrogative sentences are not considered statements in Logic.
Definition: Quantifiers
A quantifier is a symbol used to specify the quantity of elements in a domain for which a statement is true.
- Universal Quantifier (∀): “for all” or “for every”
- Existential Quantifier (∃): “there exists at least one”
Definition: Quantified Statement
A quantified statement is an open statement that becomes a definite statement when a quantifier is applied to it.
Definition: Duality
Two formulae A and B are said to be duals of each other, if either one can be obtained from the other by replacing ∧ by ∨ and ∨ by ∧.
- Replace AND (∧) with OR (∨) and OR (∨) with AND (∧)
Definition: Simple and Compound Statements
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.
Definition: Logical Equivalence
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.
Definition: Statement
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.
Definition: Truth Value of a Statement
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.
Definition: Logical Connectives
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''.
Definition: Duality
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)
-
Replacing ∨ (or) by ∧ (and)
-
Replacing T (tautology) by F (contradiction)
-
Replacing F (contradiction) by T (tautology)
while keeping negations unchanged.
Key Points
Key Points: Types of Statement
| Type | Definition |
|---|---|
| Simple Statement | Cannot be broken into smaller statements |
| Compound Statement | Formed by combining two or more simple statements |
| Open Statement | Contains variables; becomes a statement when values are assigned |
Key Points: Logical Connectives
| Type of compound statement | Connective | Symbol | Example | Set Theory Relation |
|---|---|---|---|---|
| Conjunction | and | ∧ | p and q : p ∧ q | Intersection of sets |
| Disjunction | or | ∨ | p or q : p ∨ q | Union of sets |
| Negation | not | ~ | negation p : ~ p | Complement of a set |
| Conditional or Implication | if...then | → or ⇒ | If p, then q : p → q | Subset relation |
| Biconditional or Double implication | if and only if, i.e., iff | ↔ or ⇔ | p iff q : p ↔ q | Equality of sets |
Key Points: Types of Statements
| Type | Key Point |
|---|---|
| Tautology | Statement always true |
| Contradiction (Fallacy) | Statement always false |
| Contingency | A statement is sometimes true, sometimes false |
Key Points: Negation
| Statement | Negation |
|---|---|
| \[\sim(\sim p)\] | ( p ) |
| \[\sim(p\wedge q)\] | \[\sim p\lor q\] |
| \[\sim(p\lor q)\] | \[\sim p\wedge\sim q\] |
| \[\sim(p\to q)\] | \[p\wedge\sim q\] |
| \[\sim(p\leftrightarrow q)\] | \[(p\wedge\sim q)\vee(\sim p\wedge q)\] |
| \[\sim(\forall x)\] | \[\exists x\] |
| \[\sim(\exists x)\] | \[\forall x\] |
| \[\sim(x<y)\] | \[x\geq y\] |
| \[\sim(x>y)\] | \[x\leq y\] |
Key Point: Converse, Inverse, and Contrapositive
| Statement Type | Form | Key Point | Equivalence |
|---|---|---|---|
| Original Statement | \[p\to q\] | If p then q | Equivalent to Contrapositive |
| Converse | \[q\to p\] | Interchange p and q | Equivalent to Inverse |
| Inverse | \[\sim p\to\sim q\] | Negate both p and q | Equivalent to Converse |
| Contrapositive | \[\sim q\to\sim p\] | Interchange + negate | Equivalent to Original Statement |
Key Points: Algebra of Statements
| Law | Statement(s) |
|---|---|
| Idempotent Law | \[\begin{array} {l}p\lor p\equiv p \\ p\land p\equiv p \end{array}\] |
| Commutative Law | \[\begin{aligned} & p\lor q\equiv q\lor p \\ & p\land q\equiv q\land p \end{aligned}\] |
| Associative Law | \[(p\lor q)\lor r\equiv p\lor(q\lor r)\equiv p\lor q\lor r\] \[(p\land q)\land r\equiv p\land(q\land r)\equiv p\land q\land r\] |
| Distributive Law | \[p\lor(q\land r)\equiv(p\lor q)\land(p\lor r)\] \[p\land(q\lor r)\equiv(p\land q)\lor(p\land r)\] |
| Identity Law | \[p\lor F\equiv p\] \[p\wedge T\equiv p\] \[p\lor T\equiv T\] \[p\wedge F\equiv F\] |
| Complement Law | \[\begin{array} {l}p\lor\sim p\equiv T \\ p\land\sim p\equiv F \end{array}\] |
| Absorption Law | \[\begin{array} {l}p\lor(p\land q)\equiv p \\ p\land(p\lor q)\equiv p \end{array}\] |
| De Morgan’s Law | \[\sim(p\lor q)\equiv\sim p\land\sim q\] \[\sim(p\wedge q)\equiv\sim p\vee\sim q\] |
| Conditional Law | \[p\to q\equiv\sim p\lor q\] |
| Biconditional Law | \[p\leftrightarrow q\equiv(p\to q)\land(q\to p)\]\[\equiv(\sim p\lor q)\land(\sim q\lor p)\] |
Key Points: Application of Logic to Switching Circuits
| Concept | Key Point |
|---|---|
| Series Connection (AND) | Current flows only when both switches are ON |
| Logic Form | \[p\wedge q\] |
| Parallel Connection (OR) | Current flows when any one or both switches are ON |
| Logic Form | \[p\vee q\] |
| Switch ON | Represented by p |
| Switch OFF | Represented by ~p |
| Complementary Switches | If one is ON, the other is OFF → \[S_1\equiv\sim S_2\] |
| Composite Circuits | Combine AND & OR operations (mixed circuits) |
Key Points: Logical Connectives
| 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
Key Points: Tautology, Contradiction and Contingency
| Type | Meaning |
|---|---|
| Tautology | Always True |
| Contradiction | Always False |
| Contingency | Sometimes T, Sometimes F |
Key Points: Quantifiers and Quantified Statements
| 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.
Key Points: Switching Circuits
| Circuit Type | Logical Form |
|---|---|
| Series | p ∧ q |
| Parallel | p ∨ q |
Switch ON = 1
Switch OFF = 0
Key Points: Converse, Inverse, Contrapositive
For p → q:
| Type | Form |
|---|---|
| Converse | q → p |
| Inverse | ∼p → ∼q |
| Contrapositive | ∼q → ∼p |
Key Points: Negation of Compound Statements
| 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) |
Key Points: Algebra of Statements
| 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) |
Important Questions [27]
- 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
- 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 ∧ ~q).
- Using truth table prove that ∼p ˄ q ≡ (p ˅ q) ˄ ∼p
- Write the following compound statement symbolically. Nagpur is in Maharashtra and Chennai is in Tamil Nadu.
- Using truth table, prove the following logical equivalence : (p ∧ q) → r ≡ p → (q → r)
- 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
- Discuss the statement pattern, using truth table : ~(~p ∧ ~q) v q
- 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)
- If a = {2, 3, 4, 5, 6}, Then Which of the Following is Not True?
- Without using truth tabic show that ~(p v q)v(~p ∧ q) = ~p
- Without using the truth table show that P ↔ q ≡ (p ∧ q) ∨ (~ p ∧ ~ q)
- The negation of p ∧ (q → r) is
- Without using truth table prove that (p ∧ q) ∨ (∼ p ∧ q) v (p∧ ∼ q) ≡ p ∨ q
- Find the symbolic form of the given switching circuit. Construct its switching table and interpret your result. diagram
- 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 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:
- Construct the Simplified Circuit for the Following Circuit
- Construct the Switching Circuit for the Following Statement : P V (~ P ∧ Q) V (- Q ∧ R) V ~ P
- 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:
Concepts [9]
- Statements and Truth Values in Mathematical Logic
- Logical Connectives
- Tautology, Contradiction, and Contingency
- Quantifier, Quantified and Duality Statements in Logic
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits
- Overview of Mathematical Logic
