Set theory & functions · worksheet solutions

Q1 Set membership (Page 1)

\(A = \{x\in \mathbf{Z}: x \text{ multiple of }3\},\; B = \{x\in \mathbf{Z}: x \text{ perfect square}\},\; C=\{4,5,9,10\},\)
\(D=\{2,4,11,14\},\; E=\{3,6,9\}\).
Which are true? 1. \(D\subseteq C\) 2. \(19\in B\) 3. \(E\in A\) 4. \(9\subset A\) 5. \(11\in D\)

Step 1 – Check each statement:

1. \(D\subseteq C\): D has 2,11,14; none of these are in C → False

2. \(19\in B\): perfect squares near 19 are 16,25 → 19 not a square → False

3. \(E\in A\): E is a set, ∈ relates elements to sets, not sets to sets → False

4. \(9\subset A\): 9 is an integer, ⊂ is for subsets, not elements → False (though 9∈A is true)

5. \(11\in D\): 11 is listed in D → True

✅ Only statement 5 is true: \(11\in D\).

Q2 Roster method (Page 2)

1. \(\{x\in\mathbb{R}:x^2=1\}\)
2. \(\{x\in\mathbb{Z}^+: x<1\}\)
3. \(\{x\in\mathbb{Z}:x^2=2\}\)
4. \(\{x\in\mathbb{Z}:1 < x \leq 6\}\)

Step 1 – Solve each condition:

1. \(x^2=1 \Rightarrow x=1\) or \(x=-1\) → roster \(\{-1,1\}\)

2. Positive integers start at 1, none <1 → empty set \(\emptyset\)

3. No integer satisfies \(x^2=2\) → \(\emptyset\)

4. Integers strictly greater than 1 and less than or equal to 6: \(2,3,4,5,6\) → \(\{2,3,4,5,6\}\)

✅ 1. \(\{-1,1\}\) 2. \(\emptyset\) 3. \(\emptyset\) 4. \(\{2,3,4,5,6\}\)

Q3 Equal sets (Page 3)

\(A=\{a,b,c,d\},\; B=\{d,e,a,c\},\; C=\{d,b,a,c\},\; D=\{a,a,d,e,c,e\}\)
Which sets are equal?

Step 1 – Remove duplicates and sort:

\(A = \{a,b,c,d\}\)

\(B = \{a,c,d,e\}\)

\(C = \{a,b,c,d\}\) (same as A)

\(D = \{a,c,d,e\}\) (same as B)

Step 2 – Compare: A = C, B = D

✅ \(A = C\) and \(B = D\)

Q4 Cardinality (Page 4)

1. Even positive integers less than 12
2. Letters in Arabic alphabet
3. Null set
4. \(\{x\in\mathbb{Z}: 5\le x<13\}\)

Step 1 – List each set explicitly:

1. Even positives <12: \(\{2,4,6,8,10\}\) → cardinality 5

2. Arabic alphabet has 28 letters → cardinality 28

3. Null set \(\emptyset\) has 0 elements → cardinality 0

4. Integers from 5 to 12 inclusive: \(\{5,6,7,8,9,10,11,12\}\) → count = 8

✅ 1. 5 2. 28 3. 0 4. 8

Q5 Power set (Page 5)

a) Find \(\mathcal{P}(A)\) for:
1. \(A=\{2,4,6\}\) 2. \(A=\{a,b,x,y\}\) 3. \(A=\{\}\)
b) Cardinality of \(\mathcal{P}(\{1,2,3,4,5\})\).

Step 1 – List all subsets (power set):

1. \(A=\{2,4,6\}\): subsets: ∅, {2},{4},{6},{2,4},{2,6},{4,6},{2,4,6} → 8 elements.

2. \(A=\{a,b,x,y\}\): \(2^4=16\) subsets: ∅, {a},{b},{x},{y},{a,b},{a,x},{a,y},{b,x},{b,y},{x,y},{a,b,x},{a,b,y},{a,x,y},{b,x,y},{a,b,x,y}.

3. \(A=\emptyset\): only subset ∅ → \(\{\emptyset\}\) (cardinality 1).

Step 2 – Cardinality of power set: \(|\mathcal{P}(\{1,2,3,4,5\})| = 2^5 = 32\).

✅ a1. 8 subsets (listed above); a2. 16 subsets; a3. \(\{\emptyset\}\); b) 32

Q6 Cartesian product (Page 6)

Let \(A=\{1,2\},\; B=\{3,4\},\; C=\{5,6\}\). Compute:
1. \(A\times B\) 2. \(B\times C\) 3. \(B^2 = B\times B\)

Step 1 – Form ordered pairs:

1. \(A\times B = \{(1,3),(1,4),(2,3),(2,4)\}\)

2. \(B\times C = \{(3,5),(3,6),(4,5),(4,6)\}\)

3. \(B\times B = \{(3,3),(3,4),(4,3),(4,4)\}\)

✅ \(A\times B = \{(1,3),(1,4),(2,3),(2,4)\}\); \(B\times C = \{(3,5),(3,6),(4,5),(4,6)\}\); \(B^2 = \{(3,3),(3,4),(4,3),(4,4)\}\)

Q7 Set operations (Page 7)

Let \(A=\{1,2,3,4\},\; B=\{3,4,5,6\},\; C=\{4,5,6,7\}\). Find:
1. \(A\cup B\) 2. \(A\cap C\) 3. \(B-A\) 4. \(C-B\)
5. Verify \(|A\cup B| = |A|+|B|-|A\cap B|\)

Step 1 – Perform each operation:

1. Union: all elements in A or B → \(\{1,2,3,4,5,6\}\)

2. Intersection: common to A and C → \(\{4\}\)

3. Set difference B−A: elements in B not in A → \(\{5,6\}\)

4. C−B: elements in C not in B → \(\{7\}\)

Step 2 – Verify cardinality formula: \(|A|=4,|B|=4,|A\cap B|=|\{3,4\}|=2\) → \(4+4-2=6\) = \(|A\cup B|\) ✓

✅ 1. \(\{1,2,3,4,5,6\}\); 2. \(\{4\}\); 3. \(\{5,6\}\); 4. \(\{7\}\); 5. verified

Q8 Membership table (Page 8)

Use membership table to show \(A \cup (A \cap B) = A\).

Step 1 – Build table with all possibilities:

\[ \begin{array}{cc|c|c} A & B & A\cap B & A\cup(A\cap B) \\ \hline T & T & T & T \\ T & F & F & T \\ F & T & F & F \\ F & F & F & F \end{array} \]

Step 2 – Compare with column for A: column 4 matches column A exactly → identity holds.

✅ \(A \cup (A \cap B) = A\) proved by membership table.

Q9 Function operations (Page 9)

\(f_1(x)=5x-6,\; f_2(x)=x^2+3x+10\). Find \(f_1+f_2\) and \(f_1f_2\).
Also, given \(A=\{a,b,c,d,e\},\; f(a)=2,f(b)=1,f(c)=4,f(d)=1,f(e)=1\),
find \(f(S)\) for \(S=\{b,c,d\}\).

Step 1 – Add functions: \((f_1+f_2)(x) = (5x-6)+(x^2+3x+10) = x^2+8x+4\).

Step 2 – Multiply functions: \((f_1f_2)(x) = (5x-6)(x^2+3x+10) = 5x^3+15x^2+50x -6x^2-18x-60 = 5x^3+9x^2+32x-60\).

Step 3 – Image of S: \(f(S) = \{f(b),f(c),f(d)\} = \{1,4,1\} = \{1,4\}\).

✅ \((f_1+f_2)(x)=x^2+8x+4\); \((f_1f_2)(x)=5x^3+9x^2+32x-60\); \(f(S)=\{1,4\}\)

Q10 Image, one-to-one, onto (Page 10)

3. \(S=\{-1,0,2,4,7\},\; f(x)=2x+1\). Find \(f(S)\).
4. \(f(a)=4,f(b)=5,f(c)=1,f(d)=3\) from \(\{a,b,c,d\}\) to \(\{1,2,3,4,5\}\) — one-to-one?
5. \(f(a)=3,f(b)=2,f(c)=1,f(d)=3\) to \(\{1,2,3\}\) — onto?

Step 1 – Compute f(S): \(f(-1)=-1,\; f(0)=1,\; f(2)=5,\; f(4)=9,\; f(7)=15\) → \(\{-1,1,5,9,15\}\).

Step 2 – Check one-to-one (injective): all outputs 4,5,1,3 are distinct → yes, one-to-one.

Step 3 – Check onto (surjective): codomain \(\{1,2,3\}\); outputs are 3,2,1,3 → all values 1,2,3 appear → onto.

✅ 3. \(f(S)=\{-1,1,5,9,15\}\); 4. Yes, one-to-one; 5. Yes, onto.

Q11 Invertible function (Page 11)

\(f:\{a,b,c\}\to\{1,2,3\},\; f(a)=2,f(b)=3,f(c)=1\).
1. Is \(f\) invertible?
2. If yes, find \(f^{-1}\).

Step 1 – Check bijection: all outputs distinct (injective) and cover {1,2,3} (surjective) → bijective → invertible.

Step 2 – Invert: \(f^{-1}(2)=a,\; f^{-1}(3)=b,\; f^{-1}(1)=c\).

✅ Yes, invertible. \(f^{-1}=\{(2,a),(3,b),(1,c)\}\)

Q12 Composition (Page 12)

\(f(x)=x^2+1,\; g(x)=2x+4\). Find \((f\circ g)(x)\).

Step 1 – Apply definition: \((f\circ g)(x)=f(g(x)) = f(2x+4)\).

Step 2 – Substitute into f: \((2x+4)^2+1 = 4x^2+16x+16+1 = 4x^2+16x+17\).

✅ \((f\circ g)(x) = 4x^2+16x+17\)

Sets & functions · worksheet Q&A