SET— Grade X

Grade X

SET || समुह

SET समुह

Math — Grade X

📚 # SET

📝 # Subtopics

🎯 TQ 1 · TM 6

🔢 K · U · A · HA

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SET

Click any topic to expand subtopics

Exercise — Model 1

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Exercise — Model 2

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Exercise — Model 3

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Exercise — Model 4

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Key Formulas

Sets & Cardinality — Grade 10

Cardinality — Union (Two Sets)

$$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$

Total elements in A or B. Subtract intersection to avoid counting shared elements twice.

Cardinality — Complement

$$n(A') = n(U) - n(A)$$

Elements NOT in A. Subtract set A from the universal set U.

Cardinality — Difference

$$n(A - B) = n(A) - n(A \cap B)$$

Elements only in A, not B. Also written $n(A_0)$. Removes the shared part.

Cardinality — Symmetric Difference

$$n(A \triangle B) = n(A) + n(B) - 2\,n(A \cap B)$$

Elements in exactly one of A or B. Equivalently $n(A_0) + n(B_0)$.

Cardinality — Universal Set

$$n(U) = n(A_0) + n(B_0) + n(A \cap B) + n\!\left(\overline{A \cup B}\right)$$

The four disjoint parts always sum to the total. Useful for finding a missing value.

Cardinality — Three Sets

$$n(A \cup B \cup C) = n(A)+n(B)+n(C) - n(A\cap B) - n(B\cap C) - n(A\cap C) + n(A\cap B\cap C)$$

Inclusion-exclusion principle for three sets. Add singles, subtract pairs, add triple back.

Set Builder — Cardinal Number

$$n(A) = \text{number of elements in } A$$

If $A = \{2, 4, 6, 8\}$ then $n(A) = 4$. Empty set: $n(\varnothing) = 0$.

Subset Count

$$\text{Number of subsets of } A = 2^{n(A)}$$

A set with $n$ elements has $2^n$ subsets including $\varnothing$ and the set itself.

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Key Concepts

Understand sets beyond memorising formulas

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What is a Set?

A set is a well-defined collection of distinct objects called elements. Written with curly braces: $A = \{1,2,3\}$. "Well-defined" means there is no ambiguity about whether an object belongs.

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Cardinality

The cardinality $n(A)$ of a set is its number of elements. Finite sets have a whole-number cardinality. The empty set has $n(\varnothing)=0$. Two sets are equinumerous if $n(A)=n(B)$.

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Four Disjoint Parts

Every Venn diagram of A and B splits into four non-overlapping regions: $A_0$ (only A), $B_0$ (only B), $A\cap B$ (both), and $(A\cup B)'$ (neither). All 16 set notations are unions of these parts.

∩

Intersection

$A \cap B$ contains elements belonging to both A and B simultaneously. If $A\cap B = \varnothing$ the sets are called disjoint. Cardinality: $n(A\cap B)$ is the overlap count.

∪

Union

$A \cup B$ contains every element in A or B or both. Never double-count: $n(A\cup B)=n(A)+n(B)-n(A\cap B)$. Union always gives a set at least as large as either original set.

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Complement

The complement $A'$ (or $\bar{A}$) contains everything in U that is NOT in A. Key identity: $A \cup A' = U$ and $A \cap A' = \varnothing$. Cardinality: $n(A')=n(U)-n(A)$.

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Set Difference

$A - B$ (also $A \setminus B$) keeps only elements of A that are not in B. It is the same as $A \cap B'$. Note: $A-B \neq B-A$ in general. Cardinality: $n(A-B)=n(A)-n(A\cap B)$.

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Symmetric Difference

$A \triangle B$ contains elements in exactly one of A or B — not both. Equivalent to $(A-B)\cup(B-A)$. Useful in "exactly one" word problems. $n(A\triangle B)=n(A)+n(B)-2n(A\cap B)$.

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Subset & Power Set

A is a subset of B ($A\subseteq B$) if every element of A is also in B. The power set $\mathcal{P}(A)$ is the set of all subsets; it has $2^{n(A)}$ elements. Every set is a subset of itself; $\varnothing$ is a subset of every set.

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Venn Diagram Strategy

Always fill in the Venn diagram from the inside out: start with $n(A\cap B)$, then find $n(A_0)=n(A)-n(A\cap B)$, then $n(B_0)=n(B)-n(A\cap B)$, and finally $n((A\cup B)')=n(U)-n(A\cup B)$.

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Word Problem Key Words

        At least one → $n(A\cup B)$  | 
        Both → $n(A\cap B)$  | 
        Only A → $n(A_0)$  | 
        Neither → $n(\overline{A\cup B})$  | 
        Exactly one → $n(A\triangle B)$.
      

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16 Set Notations Summary

Using parts p, q, r, s: $\varnothing$=0 · $A_0$=p · $B_0$=q · $A\cap B$=r · $(A\cup B)'$=s · $A$=p+r · $B$=q+r · $A\cup B$=p+q+r · $A\triangle B$=p+q · $U$=p+q+r+s.