Definitions [10]
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.
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”
A quantified statement is an open statement that becomes a definite statement when a quantifier is applied to it.
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 (∧)
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)
-
Replacing ∨ (or) by ∧ (and)
-
Replacing T (tautology) by F (contradiction)
-
Replacing F (contradiction) by T (tautology)
while keeping negations unchanged.
Key Points
| 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 |
| 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 |
| 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 | \[p\leftrightarrow q\equiv(p\to q)\land(q\to p)\]\[\equiv(\sim p\lor q)\land(\sim q\lor p)\] |
| Logical Form | Set Form |
|---|---|
| p ∧ q | A ∩ B |
| p ∨ q | A ∪ B |
| ∼p | A′ (complement) |
| p → q | A ⊂ B |
| p ↔ q | A = B |
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
| 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
Important Questions [20]
- If p : He swims q : Water is warm Give the verbal statement for the following symbolic statement: p ↔ ~ q
- If p : He swims q : Water is warm Give the verbal statement for the following symbolic statement. q → p
- Consider the following statements. If D is dog, then D is very good. If D is very good, then D is dog. If D is not very good, then D is not a dog.
- Negation of “Some men are animal” is ______.
- Converse of the statement q → p is ______.
- Evaluate: ∫ X . Log X Dx
- Write Converse, Inverse Contrnpositlve of the Statement "If Two Bi Act Arc Not Congruent Then Their Areas Are Not Equal.
- If p : He swims q : Water is warm Give the verbal statement for the following symbolic statement. ~ (p ∨ q)
- Write the converse, inverse, and contrapositive of the statement. "If 2 + 5 = 10, then 4 + 10 = 20."
- If p ∨ q is true, then the truth value of ∼ p ∧ ∼ q is ______.
- Using the Rules of Negation, Write the Negatlon of the Following:
- Write the Truth Value of the Negation of the Following Statement :
- Write the Truth Value of the Negation of the Following Statement :
- Express the Truth of Each of the Following Statements Using Venn Diagram. (1) All Teachers Are Scholars and Scholars Are Teachers. (2) If a Quadrilateral is a Rhombus Then It is a Parallelogram..
- Draw Venn Diagram for the Truth of the Following Statements : All Rational Number Are Real Numbers.
- Draw Venn Diagram for the Truth of the Following Statements : Some Rectangles Are Squares.
- Express the Truth of Each of the Following Statements Using Venn Diagrams:
- Express the Truth of the Following Statements with the Help of Venn Diagram: (A) No Circles Are Polygon (B) If a Quadrilateral is Rhombus , Then It is a Parallelogram .
- Using the Venn Diagram. Examine the Logical Equivalence of the Following Statements:
- Express the Truth of Each of the Following Statements by Venn Diagram: (A) Some Hardworking Students Are Obedient. (B) No Circles Are Polygons. (C) All Teachers Are Scholars and Scholars Are Teachers.
