Discrete Mathematics Answers

(i) Show that if A ⊆ C and B ⊆ D, then A×B ⊆ C ×D
(ii) Let A ={a , b , c ,d}and B ={y , z}. Find (a) A×B (b) B×A

What is the cardinality of each of these sets?
(a)∅
(b){∅}
(c){∅,{∅}}
(d){∅,{∅},{∅,{∅}}}

(a) Use a Venn diagram to illustrate the relationships A ⊆ B and B ⊆ C.
(b) Use a Venn diagram to illustrate the relationships A ⊂ B and B ⊂ C.
(c) Use a Venn diagram to illustrate the relationships A ⊂ B and A ⊂ C.
(d) Suppose that A , B, and C are sets such that A ⊆ B and B ⊆ C. Show that A ⊆ C.
(e) Find two sets A and B such that A ∈ B and A ⊆ B.

(i) Determine whether each of these statements is true or false.
(a) 0∈∅
(b)∅∈{0}
(c){0}⊂∅
(d)∅⊂{0}
(e){0}∈{0}
(f){0}⊂{0}
(g){∅}⊆{∅}
(ii) Determine whether these statements are true or false.
(a)∅∈{∅}
(b)∅∈{∅,{∅}}
(c){∅}∈{∅}
(d){∅}∈{{∅}}
(e){∅}⊂{∅,{∅}}
(f){{∅}}⊂{∅,{∅}}
(g){{∅}}⊂{{∅},{∅}}

For each of the following sets, determine whether 2 is an element of that set.
(a){x∈R|x is an integer greater than 1}
(b){x∈R|x is the square of an integer}
(c){2 ,{2}} (d){{2},{{2}}}
(e){{2},{2 ,{2}}} (f){{{2}}}

For each of these pairs of sets,determine whether the first is a subset of the second,the second is a subset of the first, or neither is a subset of the other.
(a) the set of airline flights from New York to New Delhi, the set of nonstop airline flights from New York to New Delhi
(b) the set of people who speak English, the set of people who speak Chinese
(c) the set of flying squirrels, the set of living creatures that can fly

List of the members of these sets.

(a){x|x is a real number such that x2 = 1.}
(b){x|x is a positive integer less than 12}
(c){x|x is the square of an integer and x < 100}
(d){x|x is an integer such that x2 = 2}

1. Check whether the set S=R - {-1} is a group under the binary operation ‘*’defined as for any two elements .
2. i. State the relation between the order of a group and the number of binary operations that can be defined on that set.
ii. How many binary operations can be defined on a set with 4 elements?
3. Discuss the group theory concept behind the Rubik’s cube.

Let A = {1,2,3,4}. Define a relation R on A by
a R b ↔ a + b ≤ 4
for every a; b ϵ A.
(a) List all the elements of R.
(b) Determine whether R has the following properties. If R has a certain property, prove this
is so, otherwise, provide a counterexample to show that it does not.
i. Reflexivity
ii. Transitivity
iii. Antisymmetry
iv. Symmetry

Let A = {1, 2, 3, 4}. Define a relation R on A by
a R b ⇐⇒ a + b ≤ 4
for every a, b ∈ A.
(a) List all the elements of R.
(b) Determine whether R has the following properties. If R has a certain property, prove this
is so, otherwise, provide a counterexample to show that it does not.
i. Reflexivity
ii. Transitivity
iii. Antisymmetry
iv. Symmetry