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Chapter 1: Problem 10

In an ordinary deck of fifty-two playing cards, for how many cards is it true a) that "This card is a ten and this card is a heart"? b) that "This card is a ten or this card is a heart"? c) that "If this card is a ten, then this card is a heart"? d) that "This card is a ten if and only if this card is a heart"?

Short Answer

Expert verified
a) 1 card, b) 16 cards, c) 49 cards, d) 1 card.

Step by step solution

01

Understand the Problem

We have a standard deck of 52 cards. We need to find out how many cards satisfy each condition given in the parts (a), (b), (c), and (d). The deck consists of 4 suits: hearts, diamonds, clubs, and spades, with each suit having 13 ranks, including a ten.
02

Solve part (a) "This card is a ten and this card is a heart"

A card must be both a ten and a heart. In a deck of cards, there is exactly one ten of hearts. So, there is 1 card that satisfies this condition.
03

Solve part (b) "This card is a ten or this card is a heart"

The phrase 'or' implies that the card can be a ten, a heart, or both. There are 4 tens in total (one in each suit) and 13 hearts (one for each rank). However, since the ten of hearts is counted in both categories, we should not double-count it. Therefore, the total number is 4 + 13 - 1 = 16.
04

Solve part (c) "If this card is a ten, then this card is a heart"

For this logical implication to be true, there cannot be tens that are not hearts. There is 1 ten of hearts and 3 other tens (clubs, diamonds, spades) that are not hearts, fulfilling the opposite of this condition. Thus, all 52 cards except these 3 satisfy the implication as the implication is vacuously true when the card is not a ten. Therefore, there are 52 - 3 = 49 cards satisfying this condition.
05

Solve part (d) "This card is a ten if and only if this card is a heart"

This biconditional condition means the card is a ten if and only if it is the ten of hearts. It cannot be any other heart or ten. Only the ten of hearts satisfies this specific dual condition, so there is 1 card satisfying this condition.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logical Implications
In discrete mathematics, logical implications are statements that suggest a cause-and-effect relationship. The implication "If this card is a ten, then this card is a heart" is of particular interest.
The structure of such an expression is often represented as "if \( p \), then \( q \)", where \( p \) is the premise or antecedent and \( q \) is the conclusion or consequent.
  • For an implication to be true, if \( p \) is true, \( q \) must also be true.
  • If \( p \) is false, the implication is vacuously true, meaning it holds true regardless of the truth of \( q \).
Since most cards aren't tens, they all vacuously uphold the implication "If this card is a ten, then this card is a heart," except for the three non-heart tens.
This is why 49 out of the 52 cards satisfy this specific logical condition.
Set Theory
Set theory is a foundational concept in discrete mathematics that helps us understand collections of objects. A standard 52-card deck can be thought of as a universal set \( U \).
  • Consider the set of cards that are tens, \( A \), and the set of cards that are hearts, \( B \).
  • Set \( A \) contains 4 elements (one ten from each suit).
  • Set \( B \) consists of 13 elements (each heart).
To illustrate part (b), "This card is a ten or a heart," we consider the union \( A \cup B \). This set encompasses all cards that are either tens or hearts or both.
Use the principle of inclusion-exclusion to prevent double-counting. This is done by subtracting the ten of hearts from the sum of the sizes of \( A \) and \( B \) to arrive at 16 cards that fulfill the condition.
Combinatorics
Combinatorics provides tools to count and combine different possibilities. It comes into play when figuring out how many cards satisfy a given condition.
For part (a), we use the combination logic to count specific cases like "This card is a ten and a heart".
  • "And" in this context means intersecting sets, or finding a card that meets both criteria. The intersection set consists of exactly 1 card鈥攖he ten of hearts.
Combinatorial logic also applies in part (b) through inclusion-exclusion, where we manage overlaps between different conditions, ensuring that a card like the ten of hearts isn't counted twice.
Effective use of combinatorics in these scenarios helps provide clear and precise answers based on the given logical expressions.
Biconditional Statements
Biconditional statements are expressions of logical equivalence, stating that two conditions are true simultaneously.
The condition "This card is a ten if and only if this card is a heart" can be written as \( p \iff q \).
  • This expression holds true only if both parts are either true or false concurrently.
For instance, only the ten of hearts meets both being a ten and being a heart simultaneously.
This is why the biconditional statement is fulfilled by only 1 card in the deck.
With biconditional logic, you can determine situations where conditions either both apply or both don't. Understanding these subtleties is crucial for solving related mathematical problems.

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Most popular questions from this chapter

Use truth tables to show that each of the following propositions is logically equivalent to \(p \leftrightarrow q\) a) \((p \rightarrow q) \wedge(q \rightarrow p)\) b) \((\neg p) \leftrightarrow(\neg q)\) c) \((p \rightarrow q) \wedge((\neg p) \rightarrow(\neg q))\) d) \(\neg(p \oplus q)\)

Prove that for all integers \(n\), if \(n\) is odd then \(n^{2}\) is odd.

Use induction to prove that the Hanoi subroutine uses \(2^{n}-1\) moves to solve the Towers of Hanoi problem for \(n\) disks. (There is a story that goes along with the Towers of Hanoi problem. It is said that on the day the world was created, a group of monks in Hanoi were set the task of solving the problem for 64 disks. They can move just one disk each day. On the day the problem is solved, the world will end. However, we shouldn't worry too much, since \(2^{64}-1\) days is a very long time - about 50 million billion years.)

Show that every compound proposition which is not a contradiction is equivalent to a proposition in disjunctive normal form. (Note: We can eliminate the restriction that the compound proposition is not a contradiction by agreeing that "F" counts as a proposition in disjunctive normal form. \(\mathbb{F}\) is logically equivalent to any contradiction.)

Simplify each of the following propositions. In your answer, the \(\neg\) operator should be applied only to individual predicates. a) \(\neg \forall x(\neg P(x))\) b) \(\neg \exists x(P(x) \wedge Q(x))\) c) \(\neg \forall z(P(z) \rightarrow Q(z))\) d) \(\neg((\forall x P(x)) \wedge \forall y(Q(y)))\) e) \(\neg \forall x \exists y P(x, y)\) f) \(\neg \exists x(R(x) \wedge \forall y S(x, y))\) \(\mathrm{g}) \neg \exists y(P(y) \leftrightarrow Q(y))\) \(\mathbf{h}) \neg(\forall x(P(x) \rightarrow(\exists y Q(x, y))))\)
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