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logo1 Definition Formalization Related Compound Statements Examples

Implications

Bernd Schr¨

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Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 1. Ideally, every theorem in mathematics and every valid

inference is of the form

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 1. Ideally, every theorem in mathematics and every valid

inference is of the form “If

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 1. Ideally, every theorem in mathematics and every valid

inference is of the form “If hypothesis

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 1. Ideally, every theorem in mathematics and every valid

inference is of the form “If hypothesis, then

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 1. Ideally, every theorem in mathematics and every valid

inference is of the form “If hypothesis, then conclusion.”

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 1. Ideally, every theorem in mathematics and every valid

inference is of the form “If hypothesis, then conclusion.”

• 2. But we must be careful: There are three entities that have

truth values associated with them:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 1. Ideally, every theorem in mathematics and every valid

inference is of the form “If hypothesis, then conclusion.”

• 2. But we must be careful: There are three entities that have

truth values associated with them: The hypothesis

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 1. Ideally, every theorem in mathematics and every valid

inference is of the form “If hypothesis, then conclusion.”

• 2. But we must be careful: There are three entities that have

truth values associated with them: The hypothesis, the conclusion

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 1. Ideally, every theorem in mathematics and every valid

inference is of the form “If hypothesis, then conclusion.”

• 2. But we must be careful: There are three entities that have

truth values associated with them: The hypothesis, the conclusion and the implication (the “if-then” statement) itself.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 1. Ideally, every theorem in mathematics and every valid

inference is of the form “If hypothesis, then conclusion.”

• 2. But we must be careful: There are three entities that have

truth values associated with them: The hypothesis, the conclusion and the implication (the “if-then” statement) itself.

• 3. At this stage, we must focus on the implication, that is, on

the connection between hypothesis and conclusion

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 1. Ideally, every theorem in mathematics and every valid

inference is of the form “If hypothesis, then conclusion.”

• 2. But we must be careful: There are three entities that have

truth values associated with them: The hypothesis, the conclusion and the implication (the “if-then” statement) itself.

• 3. At this stage, we must focus on the implication, that is, on

the connection between hypothesis and conclusion, not on the hypothesis and conclusion themselves.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 4. True statements must imply true statements.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 4. True statements must imply true statements. For example,

“If n is a prime number that is greater than 2, then n is

• dd” is a true statement.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 4. True statements must imply true statements. For example,

“If n is a prime number that is greater than 2, then n is

• dd” is a true statement.
• 5. Ideally, we never slip away from dealing with true

statements.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 4. True statements must imply true statements. For example,

“If n is a prime number that is greater than 2, then n is

• dd” is a true statement.
• 5. Ideally, we never slip away from dealing with true
• statements. But sometimes a false hypothesis is used

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 4. True statements must imply true statements. For example,

“If n is a prime number that is greater than 2, then n is

• dd” is a true statement.
• 5. Ideally, we never slip away from dealing with true
• statements. But sometimes a false hypothesis is used,

inadvertently or deliberately.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 4. True statements must imply true statements. For example,

“If n is a prime number that is greater than 2, then n is

• dd” is a true statement.
• 5. Ideally, we never slip away from dealing with true
• statements. But sometimes a false hypothesis is used,

inadvertently or deliberately.

• 6. Logic must be robust enough to stay internally consistent.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 4. True statements must imply true statements. For example,

“If n is a prime number that is greater than 2, then n is

• dd” is a true statement.
• 5. Ideally, we never slip away from dealing with true
• statements. But sometimes a false hypothesis is used,

inadvertently or deliberately.

• 6. Logic must be robust enough to stay internally consistent.
• 7. In the following examples, we illustrate what false

hypotheses can do. The following examples are not useful mathematics.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 8. False statements can imply false statements.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 8. False statements can imply false statements. For example,

if we assume 1 > 2, then we can derive −1 > 0 by subtracting 2 on both sides.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 8. False statements can imply false statements. For example,

if we assume 1 > 2, then we can derive −1 > 0 by subtracting 2 on both sides.

• 9. False statements can imply true statements.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 8. False statements can imply false statements. For example,

if we assume 1 > 2, then we can derive −1 > 0 by subtracting 2 on both sides.

• 9. False statements can imply true statements. For example,

if we assume 1 > 2, we can first derive −1 > 0 and then, because multiplication with a number that is > 0 preserves the inequality, we can infer 1 > 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 8. False statements can imply false statements. For example,

if we assume 1 > 2, then we can derive −1 > 0 by subtracting 2 on both sides.

• 9. False statements can imply true statements. For example,

if we assume 1 > 2, we can first derive −1 > 0 and then, because multiplication with a number that is > 0 preserves the inequality, we can infer 1 > 0.

• 10. To overcome the rather funny feeling that the above

arguments may leave:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 8. False statements can imply false statements. For example,

if we assume 1 > 2, then we can derive −1 > 0 by subtracting 2 on both sides.

• 9. False statements can imply true statements. For example,

if we assume 1 > 2, we can first derive −1 > 0 and then, because multiplication with a number that is > 0 preserves the inequality, we can infer 1 > 0.

• 10. To overcome the rather funny feeling that the above

arguments may leave: Realize that we focus on nothing but internal validity of the argument.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 8. False statements can imply false statements. For example,

if we assume 1 > 2, then we can derive −1 > 0 by subtracting 2 on both sides.

• 9. False statements can imply true statements. For example,

if we assume 1 > 2, we can first derive −1 > 0 and then, because multiplication with a number that is > 0 preserves the inequality, we can infer 1 > 0.

• 10. To overcome the rather funny feeling that the above

arguments may leave: Realize that we focus on nothing but internal validity of the argument. This focus will be important for doing proofs by contradiction.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Which Implications are True?

• 8. False statements can imply false statements. For example,

if we assume 1 > 2, then we can derive −1 > 0 by subtracting 2 on both sides.

• 9. False statements can imply true statements. For example,

if we assume 1 > 2, we can first derive −1 > 0 and then, because multiplication with a number that is > 0 preserves the inequality, we can infer 1 > 0.

• 10. To overcome the rather funny feeling that the above

arguments may leave: Realize that we focus on nothing but internal validity of the argument. This focus will be important for doing proofs by contradiction.

• 11. The only thing that definitely must not be allowed is for a

true statement to imply a false statement.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Implications/Conditional Statements

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Implications/Conditional Statements

• Definition. Let p and q be propositions. Then the statement “p

implies q” (or, “if p, then q”) is false if and only if p is true and q is false.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

• 1. Statements symbolized by a single letter are called

primitive propositions.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

• 1. Statements symbolized by a single letter are called

primitive propositions.

• 2. A symbol ◦ that can be put between two primitive

propositions p and q so that p◦q is a statement is called a connective.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

• 1. Statements symbolized by a single letter are called

primitive propositions.

• 2. A symbol ◦ that can be put between two primitive

propositions p and q so that p◦q is a statement is called a connective.

• 3. A statement that is constructed from other statements using

connectives is called a compound statement.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

• 1. Statements symbolized by a single letter are called

primitive propositions.

• 2. A symbol ◦ that can be put between two primitive

propositions p and q so that p◦q is a statement is called a connective.

• 3. A statement that is constructed from other statements using

connectives is called a compound statement.

• 4. Because compound statements are very similar to

functions of several variables, we will also sometimes call them logical functions.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

p q p ⇒ q

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

p q p ⇒ q FALSE

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

p q p ⇒ q FALSE FALSE

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

p q p ⇒ q FALSE FALSE FALSE

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

p q p ⇒ q FALSE FALSE FALSE TRUE

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

p q p ⇒ q FALSE FALSE FALSE TRUE TRUE

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

p q p ⇒ q FALSE FALSE FALSE TRUE TRUE FALSE

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

p q p ⇒ q FALSE FALSE FALSE TRUE TRUE FALSE TRUE

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

p q p ⇒ q FALSE FALSE FALSE TRUE TRUE FALSE TRUE TRUE

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

p q p ⇒ q FALSE FALSE TRUE FALSE TRUE TRUE FALSE TRUE TRUE

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

p q p ⇒ q FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

p q p ⇒ q FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE FALSE TRUE TRUE

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

The Truth Table of p ⇒ q

p q p ⇒ q FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE FALSE TRUE TRUE TRUE

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Equivalence and Converse

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Equivalence and Converse

• 1. Let p and q be two primitive propositions. Then the

compound statement p ⇔ q (read “p if and only if q” or “p iff q”) is true if and only if p and q have the same truth value.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Equivalence and Converse

• 1. Let p and q be two primitive propositions. Then the

compound statement p ⇔ q (read “p if and only if q” or “p iff q”) is true if and only if p and q have the same truth value.

• 2. Such a statement is also called a biconditional or an

equivalence.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Equivalence and Converse

• 1. Let p and q be two primitive propositions. Then the

compound statement p ⇔ q (read “p if and only if q” or “p iff q”) is true if and only if p and q have the same truth value.

• 2. Such a statement is also called a biconditional or an

equivalence.

• 3. The equivalence p ⇔ q is true if and only if the implication

p ⇒ q is true and the implication q ⇒ p is true.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Equivalence and Converse

• 1. Let p and q be two primitive propositions. Then the

compound statement p ⇔ q (read “p if and only if q” or “p iff q”) is true if and only if p and q have the same truth value.

• 2. Such a statement is also called a biconditional or an

equivalence.

• 3. The equivalence p ⇔ q is true if and only if the implication

p ⇒ q is true and the implication q ⇒ p is true.

• 4. The implication q ⇒ p is called the converse of the

implication p ⇒ q.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

Equivalence and Converse

• 1. Let p and q be two primitive propositions. Then the

compound statement p ⇔ q (read “p if and only if q” or “p iff q”) is true if and only if p and q have the same truth value.

• 2. Such a statement is also called a biconditional or an

equivalence.

• 3. The equivalence p ⇔ q is true if and only if the implication

p ⇒ q is true and the implication q ⇒ p is true.

• 4. The implication q ⇒ p is called the converse of the

implication p ⇒ q.

• 5. Truth of an implication does not imply truth of its converse

and vice versa.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

For each compound statement below, determine if it is true. Then state the converse and determine if the converse is true.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

For each compound statement below, determine if it is true. Then state the converse and determine if the converse is true.

• 1. If n is a prime number that is greater than 2, then n is odd.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

For each compound statement below, determine if it is true. Then state the converse and determine if the converse is true.

• 1. If n is a prime number that is greater than 2, then n is odd.

(True, converse is false.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

For each compound statement below, determine if it is true. Then state the converse and determine if the converse is true.

• 1. If n is a prime number that is greater than 2, then n is odd.

(True, converse is false.)

• 2. If n and n+2 are twin prime numbers greater than 3, then

n+1 is divisible by 3.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

For each compound statement below, determine if it is true. Then state the converse and determine if the converse is true.

• 1. If n is a prime number that is greater than 2, then n is odd.

(True, converse is false.)

• 2. If n and n+2 are twin prime numbers greater than 3, then

n+1 is divisible by 3. (True, converse is false.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

For each compound statement below, determine if it is true. Then state the converse and determine if the converse is true.

• 1. If n is a prime number that is greater than 2, then n is odd.

(True, converse is false.)

• 2. If n and n+2 are twin prime numbers greater than 3, then

n+1 is divisible by 3. (True, converse is false.)

• 3. If our goalkeeper does his job, then we will win the game.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

For each compound statement below, determine if it is true. Then state the converse and determine if the converse is true.

• 1. If n is a prime number that is greater than 2, then n is odd.

(True, converse is false.)

• 2. If n and n+2 are twin prime numbers greater than 3, then

n+1 is divisible by 3. (True, converse is false.)

• 3. If our goalkeeper does his job, then we will win the game.

(False, converse is false.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

For each compound statement below, determine if it is true. Then state the converse and determine if the converse is true.

• 1. If n is a prime number that is greater than 2, then n is odd.

(True, converse is false.)

• 2. If n and n+2 are twin prime numbers greater than 3, then

n+1 is divisible by 3. (True, converse is false.)

• 3. If our goalkeeper does his job, then we will win the game.

(False, converse is false.)

• 4. If n = 2, then there are positive integers a,b,c so that

an +bn = cn.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

For each compound statement below, determine if it is true. Then state the converse and determine if the converse is true.

• 1. If n is a prime number that is greater than 2, then n is odd.

(True, converse is false.)

• 2. If n and n+2 are twin prime numbers greater than 3, then

n+1 is divisible by 3. (True, converse is false.)

• 3. If our goalkeeper does his job, then we will win the game.

(False, converse is false.)

• 4. If n = 2, then there are positive integers a,b,c so that

an +bn = cn. (True, converse is true.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

For each compound statement below, determine if it is true. Then state the converse and determine if the converse is true.

• 1. If n is a prime number that is greater than 2, then n is odd.

(True, converse is false.)

• 2. If n and n+2 are twin prime numbers greater than 3, then

n+1 is divisible by 3. (True, converse is false.)

• 3. If our goalkeeper does his job, then we will win the game.

(False, converse is false.)

• 4. If n = 2, then there are positive integers a,b,c so that

an +bn = cn. (True, converse is true.)

• 5. If f is continuous, then f is differentiable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications

logo1 Definition Formalization Related Compound Statements Examples

For each compound statement below, determine if it is true. Then state the converse and determine if the converse is true.

• 1. If n is a prime number that is greater than 2, then n is odd.

(True, converse is false.)

• 2. If n and n+2 are twin prime numbers greater than 3, then

n+1 is divisible by 3. (True, converse is false.)

• 3. If our goalkeeper does his job, then we will win the game.

(False, converse is false.)

• 4. If n = 2, then there are positive integers a,b,c so that

an +bn = cn. (True, converse is true.)

• 5. If f is continuous, then f is differentiable. (False, converse

is true.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Implications