TN 12th Maths — Learn, Visualize, Practice

1. Binary operations

Addition, subtraction and multiplication all share a common shape: you feed in two elements of a set and get back a single element. Discrete mathematics makes this idea precise with the notion of a binary operation.

DefinitionBinary operation

A binary operation $*$ on a non-empty set $S$ is a function $* : S \times S \to S$ . That is, to every ordered pair $(a, b)$ with $a, b \in S$ it assigns exactly one element $a * b$ , and crucially that result again lies in $S$ .

The requirement that $a * b$ stays inside $S$ is called closure. It is built into the definition — if the output ever escapes $S$ , then $*$ is not a binary operation on $S$ .

Familiar examples

Addition $+$ and multiplication $\times$ are binary operations on $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$ .
Subtraction $-$ is a binary operation on $\mathbb{Z}$ , but not on $\mathbb{N}$ (for example $2 - 5 = -3 \notin \mathbb{N}$ ).
Matrix multiplication is a binary operation on the set of all $2 \times 2$ real matrices.
Union $\cup$ and intersection $\cap$ are binary operations on the power set of any set.

ExampleTesting closure

Is

a * b = a + b - ab

a binary operation on the set of integers

\mathbb{Z}

?

2. Properties of binary operations

Once we have a binary operation $*$ on a set $S$ , we can ask what algebraic laws it obeys. Four properties matter most.

DefinitionCommutative

$*$ is commutative if $a * b = b * a$ for all $a, b \in S$ .

DefinitionAssociative

$*$ is associative if $(a * b) * c = a * (b * c)$ for all $a, b, c \in S$ .

DefinitionIdentity element

An element $e \in S$ is an identity for $*$ if $a * e = e * a = a$ for every $a \in S$ . For example $0$ is the identity for addition and $1$ is the identity for multiplication.

DefinitionInverse element

If an identity $e$ exists, then $b \in S$ is an inverse of $a$ when $a * b = b * a = e$ . Under addition the inverse of $a$ is $-a$ ; under multiplication the inverse of $a \ne 0$ is $\tfrac{1}{a}$ .

Note: Commutativity and associativity are different questions. Subtraction on $\mathbb{Z}$ is neither: $3 - 5 \ne 5 - 3$, and $(8 - 3) - 2 = 3$ while $8 - (3 - 2) = 7$.

ExampleChecking associativity

On

\mathbb{R}

define

a * b = a + b + ab

. Is

*

associative?

3. Mathematical logic: statements

Logic studies reasoning. Its atoms are statements (also called propositions).

DefinitionStatement (proposition)

A statement is a declarative sentence that is either true or false, but not both. Its truth value is $T$ (true) or $F$ (false).

Statement or not?

" $2 + 3 = 5$ " is a statement (true). " $7$ is even" is a statement (false).
"Chennai is in Tamil Nadu" is a statement (true).
"Close the door." (a command), "What time is it?" (a question) and " $x + 1 = 4$ " (depends on $x$ ) are NOT statements.

A statement built from a single idea is a simple statement. Combining simple statements with connective words like and, or, not gives a compound statement. We usually name simple statements with letters $p, q, r, \dots$

4. Logical connectives and their truth tables

Five connectives let us build every compound statement. Each is defined completely by a truth table listing the output for every combination of inputs.

Negation $\neg p$ ("not $p$ ") — reverses the truth value.
Conjunction $p \wedge q$ (" $p$ and $q$ ") — true only when both are true.
Disjunction $p \vee q$ (" $p$ or $q$ ") — false only when both are false (inclusive or).
Conditional $p \to q$ ("if $p$ then $q$ ") — false only when $p$ is true and $q$ is false.
Biconditional $p \leftrightarrow q$ (" $p$ if and only if $q$ ") — true exactly when $p$ and $q$ have the same truth value.

Watch out: The conditional $p \to q$ trips students up: whenever the hypothesis $p$ is false, the whole statement is automatically true ("vacuously true"). Only a true $p$ with a false $q$ makes $p \to q$ false.

Key idea: Open the Visualize tab to build these truth tables yourself — pick any compound expression and watch the result column fill in row by row.

In the conditional $p \to q$ , the statement $q \to p$ is its converse, $\neg p \to \neg q$ its inverse, and $\neg q \to \neg p$ its contrapositive.

5. Tautology, contradiction, contingency

Look at the final column of a compound statement's truth table. Three cases can occur.

DefinitionTautology

A compound statement that is true for every possible assignment of truth values to its components. Example: $p \vee \neg p$ .

DefinitionContradiction

A compound statement that is false for every assignment. Example: $p \wedge \neg p$ .

DefinitionContingency

A compound statement that is true for some assignments and false for others — neither a tautology nor a contradiction. Example: $p \wedge q$ .

ExampleIdentifying a tautology

Show that

p \vee \neg p

is a tautology.

6. Logical equivalences

DefinitionLogically equivalent

Two statements are logically equivalent, written $\equiv$ , when they have identical truth values in every row of the truth table. Equivalently, $A \equiv B$ means $A \leftrightarrow B$ is a tautology.

The most useful equivalences to memorise:

De Morgan's laws

\neg(p \wedge q) \equiv \neg p \vee \neg q, \qquad \neg(p \vee q) \equiv \neg p \wedge \neg q

Conditional as disjunction

p \to q \equiv \neg p \vee q

Contrapositive

p \to q \equiv \neg q \to \neg p

Note: De Morgan's laws say "not (both)" is the same as "at least one not", and "not (either)" is the same as "neither". They turn a negation of a compound statement into a compound of negations.

ExampleVerifying a De Morgan law

Verify

\neg(p \wedge q) \equiv \neg p \vee \neg q

using a truth table.

7. The duality principle

Many logical equivalences come in mirror-image pairs. The duality principle explains why.

DefinitionDual of a statement

The dual of a compound statement is obtained by interchanging $\wedge$ with $\vee$ (and swapping the constants true $\mathbf{T}$ and false $\mathbf{F}$ ), leaving negations and the variables alone.

Principle: if two statements are logically equivalent, then so are their duals. This is exactly why De Morgan's two laws, and the two distributive laws, always appear together — each is the dual of the other.

Dual pairs

The dual of $p \wedge (q \vee r)$ is $p \vee (q \wedge r)$ .
The dual of $p \vee \mathbf{F} \equiv p$ is $p \wedge \mathbf{T} \equiv p$ .