universal and existential quantifiers examples pdf

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Universal Quantiﬁer Existential Quantiﬁer Mixing Quantiﬁers Binding Variables Negation Logic Programming Transcribing English into Logic Further Examples & Exercises Quantiﬁers Introduction A predicate becomes a proposition when we assign it ﬁxed values. H�|SMs� �^�+�f"�B����g����c5Xx\$9����ӷ=�^lo�}˾hC|+?��,#�rR��s�ܩ��}Qak��?Tp�Ӗ��-1Пŏ��E��b��I�s�P. ?�r��W(����8Fa2|f�H�;w�S�\2Ic2=�eH�_�#��'��0v����t���t�?��r=��Q��ׅ�7m2@ieEwB�vL����'*�V+:�����>@.22.�#��e*�\$7��>0-�TYMx����y�a�/ۥ�(�Fv ����:�! Someone is loved by everyone t m has(m,t) Every monkey shares a tail!

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In Fact, there is no limitation on the number of different quantifiers that can be defined, such as “exactly two”, “there are no more than three”, “there are at least 10”, and so on. 0000002057 00000 n • Quantifiers, universal, existential statements, universal conditional statements • Reading & writing quantified statements • Negation of quantified statements • Converse, Inverse and contrapositive of universal conditional statements • Statements with multiple quantifiers • Argument with quantified statements. 0000001267 00000 n 0000005058 00000 n § 11.2 Mixed quantifiers We now consider sentences with multiple quantifiers in which the quantifiers are “mixed”—some universal and some existential. stream <>

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0000010208 00000 n Chapter 12: Methods of Proof for Quantifiers § 12.1 Valid quantifier steps The two simplest rules are the elimination rule for the universal quantifier and the introduction rule for the existential quantifier.

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0000004366 00000 n H�lSMo�0��+�hK��1`��H�*Ej�K�6�"��l�������BZUHx̌߼�\$=>(����RP?&�+[@k}��&���6�BJM%m��PP? 0000005854 00000 n �P&��".

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0000010870 00000 n 0000009579 00000 n "n�l.��5�h�k�#2|�� k�@ wo��@����V6�q�X�@x�'2�*TX�0�Q��dyU�e�ZD%[�����?�^ If n= 2, we obtain the proposition 2 is an odd integer (False) A propositional function is true or false of an individual. 0000005079 00000 n 0000005726 00000 n Examples • ‘For all x ∈ R, there exists y ∈ R such that x+ y = 4.’ This statement says that the following in this exact order: 1. Although the universal and existential quantifiers are the most important in Mathematics and Computer Science, they are not the only ones.

<> Universal elimination This rule is sometimes called universal instantiation. 0000001862 00000 n 0000002451 00000 n After x is set, we can ﬁnd at AT LEAST ONE y based on x such that x +y = 4.

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However, another way to make a predicate into a proposition is to quantify it. 0000010229 00000 n

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Notation: universal quantifier ∀ xP (x) ‘For all x, P(x)’, ‘For every x, P(x)’ The variable x is bound by the universal quantifier producing a proposition. endobj 0000003192 00000 n

•Quantifiers: Universal and Existential •Nesting of Quantifiers •Applications Rules of Inference . 0000008929 00000 n For example: p(x): x is in Texas, D= set of cities and towns in the US. 0000001087 00000 n 2 0 obj _____ Example: U={1,2,3} ∀ xP (x)⇔ P (1) ∧ P (2) ∧ P (3) • Existential P(x) is true for some x in the universe of discourse. '��j昉��ru�-����R�!����� 0000001634 00000 n Quantifiers • Universal P(x) is true for every x in the universe of discourse. ����yb �ʆ`h�`.��P� ������1;�CI��r2��i��h2�eƓ�RT5FC�R-*G�#��r1�b�xx�p4�ȊfS��o7B�:04Z3E��\`\2T(2# �� \$�b ��l�L�1���]� ��e�a�� fSq�Rh��. Example Every monkey has a tail x y P(x,y) y x P(x,y) m t has(m,t) Everybody loves somebody vs. U �ĚP.����D4O�Tب~�K�aN��T#�Cg1�5NbP�����v��\$��'{T{�w�#�+��Р��x� ��M� endstream endobj 94 0 obj 275 endobj 60 0 obj << /Type /Page /Parent 57 0 R /Resources 61 0 R /Contents [ 70 0 R 72 0 R 77 0 R 81 0 R 85 0 R 87 0 R 89 0 R 91 0 R ] /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 61 0 obj << /ProcSet [ /PDF /Text ] /Font << /F2 74 0 R /TT2 66 0 R /TT4 62 0 R /TT6 63 0 R /TT8 79 0 R /TT10 83 0 R >> /ExtGState << /GS1 92 0 R >> /ColorSpace << /Cs5 68 0 R >> >> endobj 62 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 117 /Widths [ 278 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 556 556 0 0 0 0 0 0 0 333 0 0 0 0 0 0 0 0 722 0 0 0 0 0 0 0 0 0 833 0 0 667 778 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 556 0 0 611 556 333 0 611 278 0 0 0 0 611 611 611 0 389 556 333 611 ] /Encoding /WinAnsiEncoding /BaseFont /Arial-BoldMT /FontDescriptor 64 0 R >> endobj 63 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 167 /Widths [ 250 0 0 0 0 0 0 0 0 0 0 0 0 0 250 0 500 500 500 500 500 0 0 0 0 500 333 0 0 0 0 0 0 722 0 0 0 667 0 778 0 389 0 0 0 0 0 0 611 0 0 0 667 722 722 1000 0 0 0 0 0 0 0 0 0 500 0 444 556 444 333 500 556 278 0 0 278 833 556 500 556 556 444 389 333 556 500 722 500 500 444 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500 ] /Encoding /WinAnsiEncoding /BaseFont /TimesNewRomanPS-BoldMT /FontDescriptor 67 0 R >> endobj 64 0 obj << /Type /FontDescriptor /Ascent 905 /CapHeight 0 /Descent -211 /Flags 32 /FontBBox [ -628 -376 2000 1010 ] /FontName /Arial-BoldMT /ItalicAngle 0 /StemV 133 >> endobj 65 0 obj << /Type /FontDescriptor /Ascent 891 /CapHeight 0 /Descent -216 /Flags 34 /FontBBox [ -568 -307 2000 1007 ] /FontName /TimesNewRomanPSMT /ItalicAngle 0 /StemV 0 >> endobj 66 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 169 /Widths [ 250 0 0 0 0 0 0 0 333 333 0 0 250 333 250 278 500 500 500 500 500 500 500 500 0 0 278 278 0 0 0 444 0 722 667 667 722 611 556 722 722 333 389 0 611 889 722 722 556 722 667 556 611 0 0 944 0 722 0 0 0 0 0 0 0 444 500 444 500 444 333 500 500 278 278 500 278 778 500 500 500 500 333 389 278 500 500 722 500 500 444 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 333 444 444 0 0 1000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 760 ] /Encoding /WinAnsiEncoding /BaseFont /TimesNewRomanPSMT /FontDescriptor 65 0 R >> endobj 67 0 obj << /Type /FontDescriptor /Ascent 891 /CapHeight 0 /Descent -216 /Flags 34 /FontBBox [ -558 -307 2000 1026 ] /FontName /TimesNewRomanPS-BoldMT /ItalicAngle 0 /StemV 133 >> endobj 68 0 obj [ /CalRGB << /WhitePoint [ 0.9505 1 1.089 ] /Gamma [ 2.22221 2.22221 2.22221 ] /Matrix [ 0.4124 0.2126 0.0193 0.3576 0.71519 0.1192 0.1805 0.0722 0.9505 ] >> ] endobj 69 0 obj 593 endobj 70 0 obj << /Filter /FlateDecode /Length 69 0 R >> stream

The variable x can set as ANY real number. 0000006291 00000 n

endobj •Universal and Existential Quantifiers ... Nested Quantifiers: Order matters! H�b```f``������f��ǀ |�@Q�