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Maths P.P.T. Connectives

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This document discusses logical connectives and truth functions. There are five basic connectives: conjunction, disjunction, negation, implication, and equivalence. Truth functions combine sentences in ways that the truth value of the longer sentence is determined by the truth values of the parts. Each connective is represented by a symbol and has a corresponding truth table that shows the possible truth value combinations of the statements joined by the connective.

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Maths P.P.T. Connectives
The words and phrases to form compound statements are called connectives. There are five basic connectives :- 1) Conjunction 2) Disjunction 3) Negation 4) Implication or Conditional 5) Equivalence or Biconditional
Truth Functions Declarative sentences (statements) are either true or false but not both. They cannot be neither. Whether a sentence is true or false determines its truth value. Functions take input values to unique output values . That is, the input values determine the output values. There are some ways of combining sentences into longer ones so that the truth value of the longer sentence is determined by the truth value of the parts that were combined to form it. So, well call the connectives , that combine sentences in this way that they form truth functions.
1) CONJUNCTION Symbol used :- Connective word :- and Its symbolic form :- p q And is often a truth functional connective . Inserting and between two statements gives a longer sentence , the truth of which is determined by the truth of the parts. If p and q are both true then the longer sentence is true. If either of p or q is false
p : John passes. h : John is happy This symbol replaces the word AND John passes AND John is happy p h
We can express all four possibilities for A and B in table form. Heres how to read the table. The second (horizontal) row says that when A is true and B is true then A and B is true . The bottom row says that when A is false and B is false then A and B is false. A Truth Table for And
2) DISJUNCTION Symbol used :- V Connective word :- or Its symbolic form :- p V q Examples :- Either Lenny or Manny left for Bermuda. (L=Lenny left for Bermuda. M=Manny left for Bermuda.) Translation : L V M
Truth Table for Disjunction The four possibilities For sentences A and B are represented by the left two (vertical) columns. The top row of Ts and Fs is the possibility where A is true and B is true. On that possibility A V B is true. The bottom row is the possibility where A and B are both false. In that case A V B is also false. Disjunction differs from conjunction on the middle two rows.
3) NEGATION Symbol used :- ~ Connective word :- not Its symbolic form :- ~ p Prefixing a statement with it is not the case that flips the truth value. Examples :- Manny left but Lenny did not leave. (L=Lenny left for Bermuda. M=Manny left for Bermuda.) Translation : M ~L
Truth Table for Negation The symbol to represent that truth function is ~ . The symbol is called tilde or just squiggle. Unlike the other connectives, tilde doesnt connect two sentences. It just flips the value of a single sentence.
s : John studies. John does NOT study: s This symbol negates the statement it precedes ~ ~ Example :-
4) IMPLICATION ( or CONDITIONAL ) Symbol used :- Connective word :- if........then Its symbolic form :- p q Some uses of if..then are truth functional and some are not . Consider: If you whistle loudly then the dog will come. Theres really just one scenario that shows the sentence to be false: whistle loudly and have the dog not come. If you whistle and the dog comes then the sentence was true. If you dont whistle then no matter whether the dog comes or not, you didnt show the
s : John studies. p : John passes. This symbol replaces the connective if then IF John studies THEN John passes. s p Example :-
Truth Table for Conditional Well use the arrow to represent the truth functional if...then; and well call statements formed with the arrow conditionals. The top two rows of the truth table for conditional are uncontroversial. The bottom two are less obvious.
Order Matters Notice on the table that the order of A and B matter. A true and B false yields a different value than A false and B true. So, unlike conjunction and disjunction where order doesnt matter, we have a different names for the different parts of the conditional. For A B, A is called the antecedent and B is called the consequent. There are many English expressions that can be translated as conditionals. The trick to symbolizing them correctly is to identify the antecedent and the consequent.
When using the connective The direction of the arrow is important. cause effect
6) EQUIVALENCE ( or BICONDITIONAL ) Symbol used :- Connective word :- if and only if Its symbolic form :- p q If we say A if B and A only if B or in other words A if and only if B we could translate that as (BA)&(AB).Well use the double arrow (or the triple bar ) as an abbreviation for that. Also, well call sentences formed by using the double arrow biconditionals. Sometimes you see iff between two sentences. Its not a typo. Its an abbreviation for if and only if. So A iff B would be translated as
s : John studies. h : John is happy This symbol replaces the words if and only if John is happy IF AND ONLY IF John studies. h s Example :-
Truth Table for Biconditional Think of the biconditional as saying that two sentences have matching truth value. The longer sentence is true when both parts are true or when both parts are false. Notice that order doesnt matter ; so we dont have special names for the letter that comes first. Unlike regular conditional, A B is equivalent to BA. The sentence AB is just an abbreviation for the more complicated sentence (BA)&(AB).
John is happy ONLY IF he studies. John is happy IF he studies. John is happy IF AND ONLY IF he studies. Example :-
Maths P.P.T. Connectives

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Maths P.P.T. Connectives

  • 2. The words and phrases to form compound statements are called connectives. There are five basic connectives :- 1) Conjunction 2) Disjunction 3) Negation 4) Implication or Conditional 5) Equivalence or Biconditional
  • 3. Truth Functions Declarative sentences (statements) are either true or false but not both. They cannot be neither. Whether a sentence is true or false determines its truth value. Functions take input values to unique output values . That is, the input values determine the output values. There are some ways of combining sentences into longer ones so that the truth value of the longer sentence is determined by the truth value of the parts that were combined to form it. So, well call the connectives , that combine sentences in this way that they form truth functions.
  • 4. 1) CONJUNCTION Symbol used :- Connective word :- and Its symbolic form :- p q And is often a truth functional connective . Inserting and between two statements gives a longer sentence , the truth of which is determined by the truth of the parts. If p and q are both true then the longer sentence is true. If either of p or q is false
  • 5. p : John passes. h : John is happy This symbol replaces the word AND John passes AND John is happy p h
  • 6. We can express all four possibilities for A and B in table form. Heres how to read the table. The second (horizontal) row says that when A is true and B is true then A and B is true . The bottom row says that when A is false and B is false then A and B is false. A Truth Table for And
  • 7. 2) DISJUNCTION Symbol used :- V Connective word :- or Its symbolic form :- p V q Examples :- Either Lenny or Manny left for Bermuda. (L=Lenny left for Bermuda. M=Manny left for Bermuda.) Translation : L V M
  • 8. Truth Table for Disjunction The four possibilities For sentences A and B are represented by the left two (vertical) columns. The top row of Ts and Fs is the possibility where A is true and B is true. On that possibility A V B is true. The bottom row is the possibility where A and B are both false. In that case A V B is also false. Disjunction differs from conjunction on the middle two rows.
  • 9. 3) NEGATION Symbol used :- ~ Connective word :- not Its symbolic form :- ~ p Prefixing a statement with it is not the case that flips the truth value. Examples :- Manny left but Lenny did not leave. (L=Lenny left for Bermuda. M=Manny left for Bermuda.) Translation : M ~L
  • 10. Truth Table for Negation The symbol to represent that truth function is ~ . The symbol is called tilde or just squiggle. Unlike the other connectives, tilde doesnt connect two sentences. It just flips the value of a single sentence.
  • 11. s : John studies. John does NOT study: s This symbol negates the statement it precedes ~ ~ Example :-
  • 12. 4) IMPLICATION ( or CONDITIONAL ) Symbol used :- Connective word :- if........then Its symbolic form :- p q Some uses of if..then are truth functional and some are not . Consider: If you whistle loudly then the dog will come. Theres really just one scenario that shows the sentence to be false: whistle loudly and have the dog not come. If you whistle and the dog comes then the sentence was true. If you dont whistle then no matter whether the dog comes or not, you didnt show the
  • 13. s : John studies. p : John passes. This symbol replaces the connective if then IF John studies THEN John passes. s p Example :-
  • 14. Truth Table for Conditional Well use the arrow to represent the truth functional if...then; and well call statements formed with the arrow conditionals. The top two rows of the truth table for conditional are uncontroversial. The bottom two are less obvious.
  • 15. Order Matters Notice on the table that the order of A and B matter. A true and B false yields a different value than A false and B true. So, unlike conjunction and disjunction where order doesnt matter, we have a different names for the different parts of the conditional. For A B, A is called the antecedent and B is called the consequent. There are many English expressions that can be translated as conditionals. The trick to symbolizing them correctly is to identify the antecedent and the consequent.
  • 16. When using the connective The direction of the arrow is important. cause effect
  • 17. 6) EQUIVALENCE ( or BICONDITIONAL ) Symbol used :- Connective word :- if and only if Its symbolic form :- p q If we say A if B and A only if B or in other words A if and only if B we could translate that as (BA)&(AB).Well use the double arrow (or the triple bar ) as an abbreviation for that. Also, well call sentences formed by using the double arrow biconditionals. Sometimes you see iff between two sentences. Its not a typo. Its an abbreviation for if and only if. So A iff B would be translated as
  • 18. s : John studies. h : John is happy This symbol replaces the words if and only if John is happy IF AND ONLY IF John studies. h s Example :-
  • 19. Truth Table for Biconditional Think of the biconditional as saying that two sentences have matching truth value. The longer sentence is true when both parts are true or when both parts are false. Notice that order doesnt matter ; so we dont have special names for the letter that comes first. Unlike regular conditional, A B is equivalent to BA. The sentence AB is just an abbreviation for the more complicated sentence (BA)&(AB).
  • 20. John is happy ONLY IF he studies. John is happy IF he studies. John is happy IF AND ONLY IF he studies. Example :-