Write the following sets in the roster form:
(i) \(A = \{x : x \in \mathbb{R},\; 2x + 11 = 15\}\)
(ii) \(B = \{x \mid x^2 = x,\; x \in \mathbb{R}\}\)
(iii) \(C = \{x \mid x\text{ is a positive factor of a prime number }p\}\)
(i) \(\{2\}\)
(ii) \(\{0,1\}\)
(iii) \(\{1,p\}\)
Write the following sets in the roster form:
(i) \(D = \{t \mid t^3 = t,\; t \in \mathbb{R}\}\)
(ii) \(E = \{w \mid \dfrac{w-2}{w+3} = 3,\; w \in \mathbb{R}\}\)
(iii) \(F = \{x \mid x^4 - 5x^2 + 6 = 0,\; x \in \mathbb{R}\}\)
(i) \(\{0,-1,1\}\)
(ii) \(\{-\tfrac{11}{3}\}\)
(iii) \(\{-\sqrt{3}, -\sqrt{2}, \sqrt{2}, \sqrt{3}\}\)
If \(Y = \{x \mid x\text{ is a positive factor of the number }2^p-1\}\), where \(2^p-1\) is a prime number, write \(Y\) in the roster form.
\(\{1,2,2^2,2^3,\dots,2^{p-1},(2^p-1)\}\)
State which of the following statements are true and which are false. Justify your answer.
(i) \(35 \in \{x \mid x\text{ has exactly four positive factors}\}\).
(ii) \(128 \in \{y \mid \text{the sum of all the positive factors of }y\text{ is }2y\}\).
(iii) \(3 \notin \{x \mid x^4 -5x^3 +2x^2 -112x +6 =0\}\).
(iv) \(496 \notin \{y \mid \text{the sum of all the positive factors of }y\text{ is }2y\}\).
(i) True
(ii) False
(iii) True
(iv) True
Given \(L=\{1,2,3,4\},\; M=\{3,4,5,6\}\) and \(N=\{1,3,5\}\). Verify that \(L-(M\cup N) = (L-M) \cap (L-N)\).
Both sides are equal (verification by computing both sets gives the same result).
If \(A\) and \(B\) are subsets of the universal set \(U\), then show that
(i) \(A \subset A\cup B\).
(ii) \(A \subset B \iff A\cup B = B\).
(iii) \((A\cap B) \subset A\).
Standard subset properties hold; each statement is true (prove by element-wise argument).
Given that \(\mathbb{N}=\{1,2,3,\dots,100\}\). Then write
(i) the subset of \(\mathbb{N}\) whose elements are even numbers.
(ii) the subset of \(\mathbb{N}\) whose elements are perfect square numbers.
(i) \(\{2,4,6,8,\dots,98,100\}\)
(ii) \(\{1,4,9,16,25,36,49,64,81,100\}\)
If \(X=\{1,2,3\}\), if \(n\) represents any member of \(X\), write the following sets containing all numbers represented by
(i) \(4n\)
(ii) \(n+6\)
(iii) \(\tfrac{n}{2}\)
(iv) \(n-1\)
(i) \(\{4,8,12\}\)
(ii) \(\{7,8,9\}\)
(iii) \(\{\tfrac{1}{2},1,\tfrac{3}{2}\}\)
(iv) \(\{0,1,2\}\)
If \(Y=\{1,2,3,\dots,10\}\), and \(a\) represents any element of \(Y\), write the following sets containing all the elements satisfying the given conditions.
(i) \(a\in Y\) but \(a^2 \notin Y\).
(ii) \(a+1=6,\; a\in Y\).
(iii) \(a\) is less than 6 and \(a\in Y\).
(i) \(\{4,5,6,7,8,9,10\}\)
(ii) \(\{5\}\)
(iii) \(\{1,2,3,4,5\}\)
Let \(A,B\) and \(C\) be subsets of the universal set \(U\). If \(A=\{2,4,6,8,12,20\},\; B=\{3,6,9,12,15\},\; C=\{5,10,15,20\}\) and \(U\) is the set of all whole numbers, draw a Venn diagram showing the relation of \(U,A,B\) and \(C\).
Venn diagram showing the three sets \(A,B,C\) inside \(U\) with elements placed according to membership (see provided diagram in workbook).
Let \(U\) be the set of all boys and girls in a school, \(G\) be the set of all girls in the school, \(B\) be the set of all boys in the school, and \(S\) be the set of all students in the school who take swimming. Some, but not all, students in the school take swimming. Draw a Venn diagram showing one of the possible interrelationships among sets \(U,G,B\) and \(S\).
One possible Venn diagram: \(U\) as the universal rectangle, two disjoint circles for \(B\) and \(G\) (or overlapping if some students counted in both), and \(S\) as a circle overlapping both showing swimmers among boys and girls.
For all sets \(A,B\) and \(C\), show that \((A-B) \cap (C-B) = A - (B\cup C)\).
Determine whether each of the statements in Exercises 13–17 is true or false. Justify your answer.
The equality holds (prove by element-wise argument).
For all sets \(A\) and \(B\), show whether \((A-B) \cup (A\cap B) = A\) is true or false.
True
For all sets \(A,B\) and \(C\), determine whether \(A-(B-C) = (A-B)-C\) is true or false.
False
For all sets \(A,B\) and \(C\), if \(A \subset B\), then is \(A\cap C \subset B\cap C\)?
True
For all sets \(A,B\) and \(C\), if \(A \subset B\), then is \(A\cup C \subset B\cup C\)?
True
For all sets \(A,B\) and \(C\), if \(A \subset C\) and \(B \subset C\), then is \(A\cup B \subset C\)?
True
Using properties of sets, prove the statement: For all sets \(A\) and \(B\), \(A\cup (B-A) = A\cup B\).
True; prove by showing each side contains the same elements (element-wise argument).
Using properties of sets, prove: For all sets \(A\) and \(B\), \(A - (A - B) = A\cap B\).
True; standard set identity (verify element-wise).
Using properties of sets, prove: For all sets \(A\) and \(B\), \(A - (A\cap B) = A - B\).
True; follows from definitions of set difference and intersection.
Using properties of sets, prove: For all sets \(A\) and \(B\), \((A\cup B) - B = A - B\).
True; verify by element-wise consideration.
Let \(T = \{x \mid \dfrac{x+5}{x-7} - 5 = \dfrac{4x-40}{13-x}\}\). Is \(T\) an empty set? Justify your answer.
\(T = \{10\}\).