Logic Packet Ms. Anderle

Name:______Date:______Period:_____ Logic Packet Ms. Anderle

Logic Packet

Logic is the study of ______, the process of using mathematical sentences to make decisions. A mathematical sentence is always in the form of a ______.

A statement or a closed sentence does not contain any ______. Each statement has a truth value. It is either true (T) or false (F). Examples: 1. Every week has seven days. ______

2. Wednesday follows Thursday. ______

Open sentences contain unknown pronouns (he, she, they, it) or variables. The truth value of an open sentence cannot be established until that unknown or variable is replaced. Otherwise, an open sentence has ______true value. The set of all replacements that will make an open sentence true is called the ______for that open sentence.

Label each sentence as a true closed sentence, a false closed sentence, an open sentence, or not a statement.

1. Sit down and each your dinner. ______

2. Why are you wearing a raincoat? ______

3. All squares are circles. ______

4. 6 + 2 = 5 ______

5. 4x + 1 = 13 ______

6. 8 < 4 ______

7. A triangle has three sides. ______

8. George Washington was the first president of the United States. ______

9. 2x – 5 = 3 ______Negations:

To form a negation of a statement, insert the word ______so that the original statement and its negation have ______.

STATEMENT: A parallelogram has two pairs of congruent sides. [True]

NEGATION: A parallelogram does not have two pairs of congruent sides. [False]

It is not true that a parallelogram has two pairs of [False] congruent sides.

It is not the case that parallelograms have two pairs [False] of congruent sides.

The most common way to express logical relationships is to assign letters to statements. The most common letters used are p, q, and r. For example, if p represents “The White House is in Washington, D.C.,” then we can say that p is a true statement. The shorthand symbol for a negation is ~. If p is a true statement, then ~p is false.

A double negation has the same meaning as the original statement. For example, the two phrases: p: I am hungry. ~(~p): It is not true that I am not hungry. have the same true value.

Write the negation of the statements below. State the truth value for the original statement and the negation.

1. All trapezoids have one pair of parallel sides. ______. 2. All rectangles are squares. ______. 3. All squares are rhombuses. ______. 4. The Geometry students in this class will take a Regents in June. ______.

Compound Statements:

Compound statements can be made by joining simple statements with connectives such as and, or, and if/then.

Consider the following simple statements.

It is raining outside. I stay at home.

From these two simple statements three basic compound statements can be formed.  It is raining outside and I stay at home.  It is raining outside or I stay at home.  If it is raining outside, then I stay at home.

A ______summarizes the truth values a compound statement takes on for all possible combinations of truth values of the simple statements it comprises.

Conjunction: A conjunction is a compound sentence that is formed by connecting two simple sentences using the word ______.

The symbol for a conjunction is: ______.

For a conjunction to be true, BOTH parts must be true. If one or more parts are false, then conjunction is false. p q p  q Truth Table for Conjunction: T T

F F Use the following statements to write a compound statement for each conjunction. Then find its truth value. p: January 1 is the first day of the year. q: -5 + 11 = -6 r: A triangle has three sides. 1. p and q

______2. r  p

______3. p  ~ r

______4. ~q  r

______

Disjunction: A disjunction is a compound sentence that is formed by connected two simple sentences using the word ______.

The symbol for a disjunction is: ______.

For the disjunction to be false, BOTH parts must be false. When at least one part is true, the disjunction is true.

Truth Table for Disjunction: p q p  q

Use the following statements to write a compound statement for each disjunction. Then find its truth value. p: Parallelograms have 4 sides. q: I have Geometry E period. r: A right triangle has one right angle. 5. p or not q

______6. r p

______7. ~p  ~ r

______8. ~q  ~ r

______

State the truth value for each of the given statements. 1. 2 + 4 = 6 and 9 is a prime number. ______

2. Triangles have 3 sides or squares have 3 sides. ______

3. 2 is even and 4 is odd. ______4. No triangle has 3 sides or all triangles have 3 sides. ______

5. A triangle is not a square and a rhombus is not a rectangle. ______

Conditionals: A conditional is a compound sentence that is formed by connecting two simple sentences using the words ______. The “if” part of the conditional is considered the ______. The “then” part of the conditional is considered the ______.

A conditional is represented by: ______It is read as “If p then q” or “p implies q”

The truth value of a conditional is false if and only if the hypothesis (first part) is true and the conclusion (second part) is false.

A conditional is not always written in “If…then…” form. However, they can be rephrased to “If…then…” form. Examples of Hidden Conditionals: 1. A quadratic equation is easy to solve if it is factored. If…then… form: ______

______

Hypothesis: ______

Conclusion:______2. The sides of a rhombus are congruent. If…then… form: ______

______

Hypothesis: ______

Conclusion:______3. The measure of an exterior angle of a triangle is found by adding the measure of the two nonadjacent interior angles. If…then… form: ______

______Hypothesis: ______

Conclusion:______

Truth Table for the Conditional:

p q p  q

Find the truth value of the given conditional. 1. p: Today is Friday. [true] q: Tomorrow is Saturday. [true] p q: If today is Friday, then tomorrow is Saturday. _____

2. p: Today is Friday. [true] q: Tomorrow is Monday. [false] p q: If today is Friday, then tomorrow is Monday. _____

3. p: Today is Tuesday. [false] q: Tomorrow is Saturday. [true] p q: If today is Tuesday, then tomorrow is Saturday. _____

4. p: Today is Thursday. [false] q: Tomorrow is Monday. [false] p  q: If today is Thursday, then tomorrow is Monday. _____

Forming Related Conditional Statements: By interchanging or negating both parts of a conditional statement, or by doing both, three related conditional statements can be formed.

Type of Statement Forming a Conditional from “If p, then q”

Converse

Inverse

Contrapositive

Relationship between pairs of related conditional statements:  A conditional statement and its contrapositive always agree in truth values. This is also known as being ______.

 The converse and inverse always have the same truth values. This also makes the converse and inverse ______.

Write the converse, inverse, and contrapositive of the given statement. Then state which one is logically equivalent. 1. If a parallelogram has congruent diagonals, then it is a rectangle. Converse: ______

Inverse: ______

Contrapositive: ______

Statement that is logically equivalent to the conditional: ______

2. If I am in Geometry, then I am happy. Converse: ______

Inverse: ______Contrapositive: ______

3. If today is Monday, then I am tired. Converse: ______

Inverse: ______

Contrapositive: ______

Given the true statement “If I study, then I pass the test.” Which statement must also be true? (1) I study if I pass the test. (2) If I do not study, then I do not pass the test. (3) If I do not pass the test, then I did not study. (4) If I pass the test, then I study.

Biconditional Statements:

Bi-Conditional – compound sentence formed by combining the two conditionals pq and qp under a conjunction “and.” It tells us that “p implies q and q implies p,” written symbolically as (p → q) Λ (q → p). Remember: q → p means “If q then p” or “p if q” p → q means “If p then q” or “p only if q”

We abbreviate the bi-conditional to be “p if q and p only if q, we say “p if and only if q.” Written symbolically as “p ↔ q”

Truth Table for Bi-Conditional:

p q p ↔ q