Let the universal set \mathcal{U} = \{x \in \mathbb{Z} : 10 \le x \le 99\}. Consider two s… — JAMB UTME

Let the universal set $\mathcal{U} = \{x \in \mathbb{Z} : 10 \le x \le 99\}$ . Consider two subsets $P$ and $Q$ of $\mathcal{U}$ , where $P = \{10a + b \in \mathcal{U} : a > b\}$ and $Q = \{10a + b \in \mathcal{U} : a \cdot b \text{ is odd}\}$ , with $a$ and $b$ representing the tens and units digits of $x$ respectively. What is the cardinality of the set $P \setminus Q$ ?

Cevap: 35 / 35 elements

Cevap

To evaluate

n(P \setminus Q)

, we calculate

n(P) - n(P \cap Q)

. Set

P

consists of all two-digit numbers where the tens digit exceeds the units digit, giving

1 + 2 + \dots + 9 = 45

numbers. Set

Q

consists of numbers with two odd digits. The intersection

P \cap Q

contains pairs of distinct odd digits

(a,b)

where

a > b

, which corresponds to

\binom{5}{2} = 10

pairs. Thus,

n(P \setminus Q) = 45 - 10 = 35

Adım Adım Çözüm

Calculate the cardinality of set

P

, denoted as

n(P)

n(P) = 45

For a two-digit integer

10a + b

a \in \{1, 2, \dots, 9\}

and

b \in \{0, 1, \dots, 9\}

. The condition

a > b

yields

0

valid values of

b

for

a=1

(1 choice: 10), 2 choices for

a=2

(20, 21), up to 9 choices for

a=9

(90 through 98). The total is

\sum_{a=1}^9 a = \frac{9 \times 10}{2} = 45

Calculate the cardinality of the intersection

P \cap Q

, denoted as

n(P \cap Q)

n(P \cap Q) = 10

Set

Q

contains numbers where the product

a \cdot b

is odd, which requires both

a

and

b

to be odd digits chosen from

\{1, 3, 5, 7, 9\}

. For

P \cap Q

, we need

a > b

with distinct odd digits. The number of such pairs

(a, b)

is given by choosing 2 distinct odd digits out of 5:

\binom{5}{2} = 10

Calculate the cardinality of the set difference

P \setminus Q

n(P \setminus Q) = 35

Using the identity

n(P \setminus Q) = n(P) - n(P \cap Q)

, we get

45 - 10 = 35

Anahtar Kavram

Cardinality of set difference and set-builder operations

Tahmini Süre:2m 30s

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