Some more foundation mathematics — notions from set theory: Tutorial problems

It’s a strange way of starting a lecture that I adopt..sometimes…I first give my students quizzes or exams…Here is some foundation mathematics for my deserving students and also, if any of my reader is interested:

Basic Notions from Set Theory:

Reference: Introduction to Analysis, Maxwell Rosenlicht, Dover Publications,

Dover Pub, math link: http://store.doverpublications.com/by-subject-mathematics.html

Exercises:

Question 1:

Let \Re be the set of real numbers and let the symbols <, \leq have their conventional meanings:

a) Show that \{ x \in \Re: 0 \leq x \leq 3\} \bigcap \{x \in \Re: -1 <x <1 \}=\{ x \in \Re: 0 \leq x <1\}

b) List the elements of

(\{2,3,4 \} \bigcup \{ x \in \Re: x^{2}-4x+3 = 0\}) \bigcap \{x \in \Re: -1 \leq x < 3 \}

c) Show that

(\{ x \in \Re: -2 \leq x \leq 0\} \bigcup \{ x \in \Re: 2 < x <4\}) \bigcap \{ x \in \Re: 0 \leq x \leq 3\} = \{ x \in \Re: 2 < x \leq 3\} \bigcup \{ 0\}

Question 2:

If A is a subset of the set S, prove that :

2a) (A^{'})^{'}=A

2b) A \bigcup A = A \bigcap A = A \bigcup \phi = A

2c) A \bigcap \phi = \phi

2d) A \times \phi = \phi

Question 3:

Let A, B, C be elements of a set S. Prove the following statements and illustrate them with diagrams:

(a) A^{'} \bigcup B^{'} = (A\bigcap B)^{'}… a De Morgan law. In words, it can be said that the union of two complements is the complement of the intersection of the two.

(b) A \bigcap (B \bigcup C) = (A \bigcap B) \bigcup (A \bigcap C)

(c) A \bigcup (B \bigcap C) = (A \bigcup B) \bigcap (A \bigcup C).

Question 4:

If A, B, C are sets, then prove that :

i) (A-B) \bigcap C = (A \bigcap C)-B

ii) (A \bigcup B)-(A \bigcap B) = (A-B) \bigcup (B-A)

iii) A-(B-C) = (A-B) \bigcup (A \bigcap B \bigcap C)

iv) (A-B) \times C = (A \times C) - (B \times C)

Question 5:

Let f be a non-empty set and for each i \in I, let X be a set. Prove that

(i) for any set B, we have :

B \bigcap \bigcup_{i \in I}X_{i} = \bigcup_{i \in I}(B \bigcap X_{i})

(ii) if each X_{i} is a subset of a given set S, then

(\bigcup_{i \in I}X_{i})^{'}=\bigcap_{i \in I}(X_{i})^{'}

Question 6:

Prove that if f: X \rightarrow Y, g: Y \rightarrow Z, and h: Z \rightarrow W are functions, then

h \circ (g \circ f) = (h \circ g) \circ f

Question 7:

Let f: X \rightarrow Y be a function, let A and B be subsets of X, and let C and D be subsets of Y. Prove that:

(i) f(A \bigcup B) = f(A) \bigcup f(B)

(ii) f(A \bigcap B) \subset f(A) \bigcap f(B)

(iii) f^{-1}(C \bigcup D) = f^{-1}(C) \bigcup f^{-1}(D)

(iv) f^{-1}(C \bigcap D) = f^{-1}(C) \bigcap f^{-1}(D)

(v) f^{-1}(f(A)) \supset A

(vi) f(f^{-1}(C)) \subset C

Question 8:

(a) Prove that a function f is one-to-one if and only if f^{-1}(f(A)) = A for all A \subset X.

(b) Prove that a function f is onto if and only if f(f^{-1}(C)) = C for all C \subset Y.

Cheers,

Nalin Pithwa

PS: These tutorial problems can be used for IIT JEE Maths, Pre RMO, RMO Maths etc. also.

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