Conditional and Biconditional Statements

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Questions and Answers

What is a conditional statement also known as?

An implication.

In the conditional statement $p \rightarrow q$, what is 'p' called?

Hypothesis.

Under what condition is the conditional statement $p \rightarrow q$ false?

When p is true and q is false.

How would you express the statement 'You can access the internet only if you have a password' in the form 'if p, then q'?

<p>If you access the internet, then you have a password.</p>
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What is the converse of the conditional statement: 'If it is raining, then the ground is wet'?

<p>If the ground is wet, then it is raining.</p>
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What is the contrapositive of the conditional statement: 'If it is raining, then the ground is wet'?

<p>If the ground is not wet, then it is not raining.</p>
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Write the contrapositive of the statement: 'If I study hard, then I will pass the exam.'

<p>If I do not pass the exam, then I did not study hard.</p>
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What is a biconditional statement?

<p>A statement of the form 'p if and only if q'.</p>
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Under what condition is the biconditional statement $p \leftrightarrow q$ true?

<p>When p and q have the same truth values.</p>
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How many rows are there in a truth table for a compound proposition consisting of 4 variables?

<p>16</p>
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In constructing a truth table, what is the first step after identifying the compound proposition?

<p>Look at the number of propositions (e.g. p, q, r) in the given compound proposition.</p>
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According to the precedence of logical operators, which operator is applied first?

<p>Negation.</p>
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What is a bit string?

<p>A sequence of zero or more bits.</p>
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What does the length of a bit string refer to?

<p>The number of bits in the string.</p>
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What is the bitwise OR of the bit strings 101 and 011?

<p>111</p>
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Express 'p is sufficient for q' using a conditional statement.

<p>$p \rightarrow q$</p>
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Express 'q is necessary for p' using a conditional statement.

<p>$p \rightarrow q$</p>
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Translate 'You can enter the contest only if you are over 18' into a conditional statement of the form 'if p, then q'.

<p>If you enter the contest, then you are over 18.</p>
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Given the statement 'I will eat an apple unless I am full', rewrite it in the form 'if p, then q'.

<p>If I am not full, then I will eat an apple.</p>
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What logical operator has exactly the opposite truth table as the biconditional operator?

<p>XOR.</p>
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State the inverse of: 'If it is sunny, then I will go to the park.'

<p>If it is not sunny, then I will not go to the park.</p>
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Rephrase 'A grade of A is sufficient for graduating with honors' as an 'if, then' statement.

<p>If you get a grade of A, then you will graduate with honors.</p>
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Rewrite the statement 'Having a ticket is necessary for entering the concert' in the form 'if p, then q'.

<p>If you enter the concert, then you have a ticket.</p>
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How does increasing the number of propositional variables affect the size (number of rows) of a truth table? Explain why.

<p>It doubles the size of the truth table because each additional variable doubles the possible combinations of truth values.</p>
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Determine the output of a bitwise AND operation between the bit strings 11001100 and 10101010.

<p><code>10001000</code></p>
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Define the term 'bitwise OR' and briefly explain with a simple example.

<p>A bitwise OR is a logical operation where the result is 1 if at least one of the corresponding bits is 1. Example: <code>101 OR 010 = 111</code>.</p>
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Flashcards

Proposition

A statement that asserts or denies something, and can be either true or false.

Propositional variable

A variable that represents a proposition, which can be either true or false.

Compound proposition

A proposition formed by combining one or more propositions using logical operators.

Logical operator

Operators used to combine or modify propositions (e.g., AND, OR, NOT).

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Truth value

The truth or falsity of a proposition.

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Truth Table

A table listing all possible truth values for a compound proposition.

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Conditional statement

A statement of the form “if p, then q.” Represented as p → q.

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Hypothesis

The part of a conditional statement following 'if'.

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Conclusion

The part of a conditional statement following 'then'.

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Converse

The statement formed by swapping the hypothesis and conclusion of a conditional statement.

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Inverse

The statement formed by negating both the hypothesis and conclusion of a conditional statement.

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Contrapositive

The statement formed by negating and swapping the hypothesis and conclusion in an if-then statement.

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Bi-conditional statement

Conditional statement that is true when both propositions have same truth values.

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Bit

A single 0 or 1 representing true or false in digital logic.

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Bit string

A sequence of bits used to represent data.

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Bit operations

Logical operations performed on corresponding bits of bit strings.

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