Search & Filtering
Prove that If A and B are any sets with the property that there is a one-to-one function from A to B and a one-to-one function from B to A, then A and B have the same cardinality.
Let 8Z be the set of all integers that are multiples of 8. Prove that 8Z has the same cardinality as 3Z, the set of all integers multiples of 3.
Show that the relation R on Z × Z defined by (a, b) R (c, d) if and only if a + d = b + c
is an equivalence relation.
Note: A relation on a set A is called an equivalence relation if it is reflexive, symmetric,
and transitive.
Show that the relation R on Z × Z defined by (a, b) R (c, d) if and only if a + d = b + c is an equivalence relation. Note: A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.
a) Suppose P (x, y) denotes the equation8, what will the truth values of the Propositions P (2,2), P (0,4).
The following table shows the income distribution of 600 families. Find the minimum income
of the riches 30% families. Also the limits of income of middle 50% of families, to the nearest
rupees.
Income Below
75
75-
150
150-
225
225-
300
300-
375
375-
400
400 &
above
No. of
families
69 137 225 46 88 25 10
Ans.: the richest 30 % families earns Rs. 222 and above per week , the middle 50% families
weekly income lies between 120 and 256.
How many different plates are there that involve 1, 2 or 3 letters followed by 1,
2, 3 or 4 digits?
How many 2 digit or 3-digit numbers can be formed using the digits 1, 3, 4, 5, 6,
8 and 9 if no repetition is allowed?
Write the converse, inverse, and contrapositive of the following conditional
propositions. (Hint: If applicable, write each conditional proposition in standard
form first.)
a. Rose may graduate if she has 120 hours of OJT credits.
b. A necessary condition for Bill to buy a computer is that he obtains
P20,000.
c. A sufficient condition for Katrina to take the algorithms course is that
she passes discrete mathematics.
d. The program is readable only if it is well-structured.
For the relation R = {(p,p) ,(q,p),(q,q),(r,r),(r,s),(s,s) ,(s,m) ,(m,m)}
1.Using warshall algorithm find the transitive closure R* of R
2.write matrix representation of R*
3.Check whether the relation of R* is an equivalence relation or a partial order.
A pet store keeps track of the purchases of customers over a fours hour period. The store manager classifies purchases as containing a dog product, a cat product, a fish product, or product for a different kind of pet. He found!
83 purchased a dog product
101 purchased a cat product
22 purchased a fish product
31 purchased a dog and a cat product
8 purchased a dog and a fish product
10 purchased a cat and a fish product
6 purchased a dog, a cat, and a fish product
34 purchased a product for a pet other than a dog, cat, or fish.
Draw a Venn diagram to find that:
(1) How many purchases were for a dog product only?
(ii) How many purchases were for a cat product only?
How many purchases were for a dog or a fish product?
(iv) How many purchases were there in total?
