Interactive lesson – Selftutors

Live Venn Visualizer

Select Set Operation:

A ∪ B (Union)
A ∩ B (Intersection)
A – B (Difference)
B – A (Difference)
A Δ B (Symmetric Difference)
A’ (Complement of A)
B’ (Complement of B)
(A ∪ B)’ (De Morgan 1)
(A ∩ B)’ (De Morgan 2)

Mathematical Formula

$A cup B$

Represents all elements that are in set A, in set B, or in both.

Comprehensive Theory Guide

1. Fundamentals of Sets

A set is a well-defined collection of distinct objects. The number of elements in a set is called the Order of the set.

Crucial Points for MCQs:

$0$ is neither positive nor negative. Therefore, the set of “non-negative integers” is ${0, 1, 2, …}$ which is equivalent to Whole Numbers ($W$).
The set of “non-positive integers” is ${0, -1, -2, …}$.
There is only one even prime number in the universe, which is $2$.

2. Subsets & Power Sets

If every element of set $A$ is an element of set $B$, then $A$ is a subset of $B$ ($A subseteq B$).

Proper Subset ($A subset B$): $B$ contains at least one element which is not in $A$.
Improper Subset: A set is an improper subset of itself ($A = B$).

Subsets Calculation Formula

If a set $S$ has $m$ elements ($n(S) = m$), then its Power Set $P(S)$ contains all possible subsets.

Total number of subsets = $2^m$
Number of proper subsets = $2^m – 1$
Number of non-empty subsets = $2^m – 1$

3. Disjoint vs Overlapping Sets

Disjoint Sets: If $A cap B = emptyset$. They have no common elements.
Overlapping Sets: If $A cap B neq emptyset$, but neither is a subset of the other ($A notsubset B$ and $B notsubset A$).

Note: The sets $A-B$, $B-A$, and $A cap B$ are always mutually disjoint. Try checking this by selecting these operations on the visualizer!

4. Algebra of Sets (Highly Tested)

Understanding these laws is critical for simplifying complex set expressions in exams.

Law Name	Property
Idempotent Law	$A cup A = A$ and $A cap A = A$
Commutative Law	$A cup B = B cup A$
Associative Law	$A cup (B cup C) = (A cup B) cup C$
Distributive Law	$A cup (B cap C) = (A cup B) cap (A cup C)$
De Morgan’s Laws	$(A cup B)’ = A’ cap B’$ $(A cap B)’ = A’ cup B’$

5. Logic & Propositions

Logic is the science of reasoning. There are two main types:

Inductive Logic: Drawing general conclusions from a limited number of observations (e.g., predicting sequence $2, 4, 6, …$).
Deductive Logic: Drawing conclusions from premises already believed to be true.

A Proposition (or statement) is a declarative sentence which may be true or false, but not both. Aristotelian logic deals with strictly two possibilities (True/False).

6. Logical Connectives (Truth Tables)

If a problem involves $n$ simple statements (like $p, q, r$), the truth table will have exactly $2^n$ rows.

Conjunction ($p wedge q$): Read as “p AND q”. True only if BOTH are true.
Disjunction ($p vee q$): Read as “p OR q”. True if AT LEAST ONE is true.
Implication ($p rightarrow q$): Read as “If p, then q”. False ONLY when $p$ is true but $q$ is false.
Bi-conditional ($p leftrightarrow q$): Read as “p if and only if q”. True when both $p$ and $q$ have the SAME truth value.

Conditionals Cheat Sheet:

If you are given a conditional statement $p rightarrow q$, you MUST memorize its variations:

1. Converse: $q rightarrow p$
2. Inverse: $sim p rightarrow sim q$
3. Contrapositive: $sim q rightarrow sim p$

* A statement and its contrapositive always have the exact same truth values!